{"id":38229,"date":"2018-03-07T19:32:49","date_gmt":"2018-03-07T19:32:49","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=38229"},"modified":"2026-04-29T11:28:57","modified_gmt":"2026-04-29T16:28:57","slug":"indefinite-integrals","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/indefinite-integrals\/","title":{"rendered":"Indefinite Integrals"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_GYEq2S8TcV4\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_GYEq2S8TcV4\" data-source-videoID=\"GYEq2S8TcV4\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Indefinite Integrals Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Indefinite Integrals\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_GYEq2S8TcV4:hover {cursor:pointer;} img#videoThumbnailImage_GYEq2S8TcV4 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/257-thumb-final-2.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_GYEq2S8TcV4\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_GYEq2S8TcV4\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Indefinite Integrals\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_GYEq2S8TcV4\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_GYEq2S8TcV4\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_GYEq2S8TcV4\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction 7RL_Function() {\n  var x = document.getElementById(\"7RL\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"7RL_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"7RL\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_Is_an_Antiderivative\" class=\"smooth-scroll\">What Is an Antiderivative?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Indefinite_Integral_Notation\" class=\"smooth-scroll\">Indefinite Integral Notation<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Common_Indefinite_Integrals\" class=\"smooth-scroll\">Common Indefinite Integrals<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Indefinite_Integral_Example\" class=\"smooth-scroll\">Indefinite Integral Example<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Antiderivatives_Key_Takeaways\" class=\"smooth-scroll\">Antiderivatives: Key Takeaways<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Indefinite_Integral_Practice_Questions\" class=\"smooth-scroll\">Indefinite Integral Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<h2><span id=\"What_Is_an_Antiderivative\" class=\"m-toc-anchor\"><\/span>What Is an Antiderivative?<\/h2>\n<p>\nNow that you\u2019ve got an understanding of what derivatives are and how to find them, we are going to move on to one of the last major topics of Calculus I: antiderivatives.<\/p>\n<p>As the name suggests, an antiderivative is essentially the undoing of a derivative. For example, given some function \\(f(x)\\), its antiderivative\u2014which is usually called \\(F(x)\\)\u2014is the function that, when you take its derivative, will give you \\(f(x)\\).<\/p>\n<div class=\"examplesentence\">\\(F(x)\\) is the antiderivative of \\(f(x)\\) <span style=\"white-space:nowrap\">if \\(F'(x)=f(x)\\).<\/span><\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Finding_a_Simple_Antiderivative\" class=\"m-toc-anchor\"><\/span>Finding a Simple Antiderivative<\/h3>\n<p>\nThis may sound a little funky, but let\u2019s try it out with an example. Find the antiderivative of \\(f(x)=2x\\).<\/p>\n<p>In order to find the antiderivative, we need to think of \\(2x\\) as being a derivative already. What function would give us \\(2x\\) as its derivative? We know that from experience that \\(x^{2}\\) would! Because of the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/derivative-properties-and-formulas\/\">power rule<\/a>, the derivative of \\(x^{2}\\) is \\(2x\\).<\/p>\n<p>Let\u2019s go ahead and write that \\(F(x)\\) is \\(x^{2}\\).<\/p>\n<div class=\"examplesentence\">\\(F(x)=x^{2}\\) \u2026<\/div>\n<p>\n&nbsp;<\/p>\n<h3>Why We Add &#8220;+C&#8221;<\/h3>\n<p>\nAntiderivatives are often as simple as doing a reversal of the power rule but with one catch. We said that the derivative of \\(x^{2}\\) is \\(2x\\). But what about the derivative of \\(x^{2}+1\\)? That would also make \\(2x\\). The same goes for \\(x^{2}+2\\), or \\(x^{2}+3\\), or \\(x^{2}\\) plus or minus any constant.<\/p>\n<p>Because of the constant rule, the derivative of these second terms will always be zero, and the resulting derivative comes out to \\(2x\\).<\/p>\n<div class=\"examplesentence\"><span style=\"display: block; margin-bottom: -1em\">\\(\\dfrac{d}{dx}(x^{2}+1)=2x\\)<\/span><br \/>\n<span style=\"display: block; margin-bottom: -1em\">\\(\\dfrac{d}{dx}(x^{2}+2)=2x\\)<\/span><br \/>\n<span style=\"display: block; margin-bottom: -1em\">\\(\\dfrac{d}{dx}(x^{2}+3)=2x\\)<\/span><br \/>\n<span>\\(\\dfrac{d}{dx}(x^{2}+c)=2x\\)<\/span><\/div>\n<p>\n&nbsp;<br \/>\nBecause of this, when we take an antiderivative, we must add an arbitrary constant to the answer. That way we are covered for every possible one of these solutions. In our example, we would write that \\(F(x)=x^2+C\\). That \\(C\\) is the arbitrary constant we need to protect our answer from missing solutions.<\/p>\n<p>Let\u2019s try another example. Find the antiderivative of \\(g(x)=5x^{4}+7x\\).<\/p>\n<p>This problem, like the first, requires a reversal of the power rule. Since the power rule tells us that we can find a derivative by multiplying by the power of \\(x\\) and then reducing the exponent by 1, we are going to do the opposite of these things, and in the opposite order. We need to add 1 to the exponent, and then divide by the new power.<\/p>\n<p>For the first term, \\(5x^{4}\\), we add 1 to the exponent, making \\(5x^{5}\\), then divide by the new power. Dividing by 5 gives us \\(x^{5}\\).<\/p>\n<p>For the second term, we increase the power by 1 to get \\(7x^{2}\\), then divide by the value of the new power to get \\(\\frac{7x^{2}}{2}\\).<\/p>\n<p>The antiderivative for this problem is then \\(G(x)=x^{5}+\\frac{7x^{2}}{2} + C\\). Don\u2019t forget to add the constant! We can check this answer by <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/definition-of-the-derivative\/\">taking the derivative<\/a>.<\/p>\n<p>Using the power rule, the first term would go back to \\(5x^{4}\\), the second term would go back to \\(7x\\), and the \\(C\\) would go to zero. This is exactly what we started the problem with!<\/p>\n<div class=\"examplesentence\"><span style=\"display: block; margin-bottom: -2em\">\\(G(x)=x^{5}+\\frac{7x^{2}}{2}+C\\)<\/span><br \/>\n<span>\\(G'(x)=5x^{4}+7x=g(x)\\)<\/span><\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Indefinite_Integral_Notation\" class=\"m-toc-anchor\"><\/span>Indefinite Integral Notation<\/h2>\n<p>\nAntiderivatives are more formally referred to as \u201cindefinite integrals.\u201d As we have talked about already, antiderivatives can be identified by their capital letter notation. However, moving forward, you\u2019ll probably come across integral notation more prominently. This means that instead of using the words \u201cfind the antiderivative of this function,\u201d these indefinite integral problems are shortened by instead asking for the integral of that function with respect to \\(x\\).<\/p>\n<div class=\"examplesentence\">\\(\\int f(x)\u2009dx\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe long \u201cS\u201d shape (\\(\\int\\)) is what we call the integral symbol, while the letters \\(dx\\) specify that we are taking the integral with respect to the variable \\(x\\).<\/p>\n<p>Every once in a while, you may have a problem in terms of some other variable, but they can be solved in the same way. As we have discussed, the antiderivative, or indefinite integral, of some function \\(f(x)\\) with respect to \\(x\\), is \\(F(x)\\) plus a constant \\(C\\).<\/p>\n<div class=\"examplesentence\">\\(\\int f(x)\u2009dx=F(x)+C\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Common_Indefinite_Integrals\" class=\"m-toc-anchor\"><\/span>Common Indefinite Integrals<\/h2>\n<h3><span id=\"Trigonometric_Integrals\" class=\"m-toc-anchor\"><\/span>Trigonometric Integrals<\/h3>\n<p>\nIf you haven\u2019t already, I want you to grab a pencil and some paper. I\u2019m about to share with you the integrals of several common functions, and your familiarization with these will help you later on in your assignments. Ready? Let\u2019s start with some helpful trig integrals.<\/p>\n<p>First, the integral of \\(\\sin(x)\\) is \\(-\\cos(x)+C\\). <\/p>\n<div class=\"examplesentence\">\\(\\int \\sin(x)\u2009dx=-\\cos(x)+C\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis should feel familiar because we know that the derivative of \\(-\\cos(x)\\) is \\(\\sin(x)\\). Likewise, the integral of \\(\\cos(x)\\) is familiar; we know it must be \\(\\sin(x)+C\\).<\/p>\n<div class=\"examplesentence\">\\(\\int \\cos(x)\u2009dx=\\sin(x)+C\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe next integral I want you to write down is the integral of \\(\\sec^{2}(x)\\), which is \\(\\tan(x)+C\\).<\/p>\n<div class=\"examplesentence\">\\(\\int \\sec^{2}(x)\u2009dx=\\tan(x)+C\\)<\/div>\n<p>\n&nbsp;<br \/>\nNext, the integral of \\(\\csc^{2}(x)\\) is \\(-\\cot(x)+C\\).<\/p>\n<div class=\"examplesentence\">\\(\\int \\csc^{2}(x)\u2009dx=-\\cot(x)+C\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe integral of \\(\\sec(x)\\tan(x)\\) is \\(\\sec(x)\\) plus a constant \\(C\\).<\/p>\n<div class=\"examplesentence\">\\(\\int \\sec(x)\\tan(x)\u2009dx=\\sec(x)+C\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe last trig integral I have for you is the integral of \\(\\csc(x)\\cot(x)\\), which is \\(-\\csc(x)+C\\).<\/p>\n<div class=\"examplesentence\">\\(\\int \\csc(x)\\cot(x)\u2009dx=-\\csc(x)+C\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Exponential_and_Logarithmic_Integrals\" class=\"m-toc-anchor\"><\/span>Exponential and Logarithmic Integrals<\/h3>\n<p>\nSince the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/derivatives-of-exponential-and-log-functions\/\">exponential function<\/a>  \\(e^{x}\\) is its own derivative, it should come as no surprise that it is also its own integral!<\/p>\n<div class=\"examplesentence\">\\(\\int e^{x}\u2009dx=e^{x}+C\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd finally, the integral of \\(\\frac{1}{x}\\) is \u2009\\(\\ln(x)+C\\).<\/p>\n<div class=\"examplesentence\">\\(\\int \\frac{1}{x}\u2009dx= \\ln(x)+C\\)<\/div>\n<p>\n&nbsp;<br \/>\nI know that this seems like a lot of information, but if you spend a few minutes writing each of these out a few times, you\u2019ll start to become familiar with them and memorization won\u2019t be far from reach. You can do this after you finish the video, or you can pause the video and do it now.<\/p>\n<h2><span id=\"Indefinite_Integral_Example\" class=\"m-toc-anchor\"><\/span>Indefinite Integral Example<\/h2>\n<p>\nLet\u2019s close by working through one more example. Determine the indefinite integral of \\(h(x)=8x^{3}+\\sec(x)\\tan(x)+e^{x}\\).<\/p>\n<div class=\"examplesentence\">\\(\\int (8x^{3}+\\sec(x)\\tan(x)+e^{x})\u2009dx\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Splitting_the_Integral_into_Terms\" class=\"m-toc-anchor\"><\/span>Splitting the Integral into Terms<\/h3>\n<p>\nThe sum and difference rule told us that we can take the derivative of a function one term at a time. The same rule applies to integrals! We can simplify this expression by splitting up the terms into separate integrals.<\/p>\n<div class=\"examplesentence\">\n<p class=\"longmath\" style=\"margin-bottom: 0em\">\\(\\int 8x^{3}\u2009dx+\\int \\sec(x)\\tan(x)\u2009dx+\\int e^{x}\u2009dx\\)<\/p>\n<\/div>\n<p>\n&nbsp;<br \/>\nEven though this line is a little longer than the first, we can more easily see how to handle each part now. First, \\(8x^{3}\\) can be integrated by reversing the power rule. Let\u2019s add 1 to the exponent to get \\(8x^{4}\\), then divide by the value of this new power to get \\(\\frac{8x^{4}}{4}=2x^{4}\\).<\/p>\n<p>We can always double-check this by taking the derivative again. The derivative of \\(2x^{4}\\) is indeed \\(8x^{3}\\). Don\u2019t forget to write the \u201cplus \\(C\\).\u201d Since we have three integrals in this problem, I\u2019m going to write a little \u201c1\u201d to indicate that this constant came from the first part.<\/p>\n<div class=\"examplesentence\">\\(2x^{4}+C_{1} +\\) \u2026<\/div>\n<p>\n&nbsp;<br \/>\nThe next term, \\(\\sec(x)\\tan(x)\\), is one that we discussed moments ago. Did you write it down? Its integral is \\(\\sec(x)\\) plus a constant. I\u2019ll write a little \u201c2\u201d with this \\(C\\) to show that it came from the second integral.<\/p>\n<div class=\"examplesentence\">\\(2x^{4}+C_{1}+\\sec(x)+C_{2}+\\)\u2026<\/div>\n<p>\n&nbsp;<br \/>\nThe third term should integrate quickly. As we said, the integral of \\(e^{x}\\) is \\(e^{x}\\), so let\u2019s write that plus a third constant.<\/p>\n<div class=\"examplesentence\">\\(2x^{4}+C_{1}+\\sec(x)+C_{2}+e^{x}+C_{3}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Combining_Constants\" class=\"m-toc-anchor\"><\/span>Combining Constants<\/h3>\n<p>\nNow, let\u2019s clean this solution up just a little bit. Because \\(C_{1}\\), \\(C_{2}\\), and \\(C_{3}\\) are all constants, they\u2019re all just numbers. And their sum is just going to be another number. For that reason, we can combine the \\(C\\)s into one constant together. So the final answer for this problem is:<\/p>\n<div class=\"examplesentence\">\\(2x^{4}+\\sec(x)+e^{x}+C\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Antiderivatives_Key_Takeaways\" class=\"m-toc-anchor\"><\/span>Antiderivatives: Key Takeaways<\/h2>\n<p>\nRemember, when integrating \\(x\\) to some power, you can reverse the power rule by adding 1 to the exponent and then dividing by the new power. Integrals of trig functions will require a little memorization, but you can get a good grip on them within just a few minutes of writing.<\/p>\n<p>The exponential function \\(e^x\\) is unchanged by integration, and the integral of \\(\\frac{1}{x}\\) is \\(\\ln(x)\\). And in all indefinite integral problems, it is critical to remember your \u201c\\(+C\\)\u201d!<\/p>\n<p>I hope this video was helpful. Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Indefinite_Integral_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Indefinite Integral Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nEvaluate the indefinite integral \\(\\int 365 \\, dx\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">\\(365x+c\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(365+c\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(3x+6x+5x+c\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(365x^2+c\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Remember, a basic integration rule is \\(\\int k \\, dx = kx + c\\), where \\(k\\) and \\(c\\) are constants.<\/p>\n<p>In this problem, 365 is in the place of the constant \\(k\\), so to get the solution, simply multiply it by \\(x\\) and add the integration constant \\(c\\). The result is \\(365x+c\\), which can be checked by taking its derivative:<\/p>\n<p style=\"text-align: center;\">\\(\\dfrac{d}{dx}(365x+c)=365\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nEvaluate the indefinite integral below:<\/p>\n<div class=\"yellow-math-quote\">\\(\\int 100x^{99} \\, dx\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(x^{100}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">\\(x^{100}+c\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(x^{99}+100\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(x^{101}+c\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>For this problem, the following formula is especially helpful, keeping in mind that \\(n \\neq -1\\):<\/p>\n<p style=\"text-align: center\">\\(\\int x^n \\, dx=\\dfrac{x^{n+1}}{n+1}+c\\)<\/p>\n<p>In this problem, \\(n=99\\), so the solution can be found by increasing that exponent by 1, dividing by \\(n+1=100\\), and finally adding the integration constant \\(c\\).<\/p>\n<p style=\"text-align:center; line-height: 60px\">\\(\\int 100x^{99} \\,\u2009dx=\\dfrac{100}{99+1}x^{99+1}+c\\)\\(\\:=x^{100}+c\\)<\/p>\n<p>This solution can be checked by taking its derivative:<\/p>\n<p style=\"text-align:center;\">\\(\\dfrac{d}{dx}(x^{100}+c)=100x^{99}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nEvaluate the indefinite integral below:<\/p>\n<div class=\"yellow-math-quote\">\\(\\int(x^4+x^3+x^2+x+1) dx\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\(x^5+x^4+x^3+x^2+x+c\\)<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\\(\\frac{1}{4}x^5+\\frac{1}{3}x^4+\\frac{1}{2}x^3+x^2+\\frac{1}{x}+c\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\(4x^3+3x^2+2x+1+c\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\">\\(\\frac{1}{5}x^5+\\frac{1}{4}x^4+\\frac{1}{3}x^3+\\frac{1}{2}x^2+x+c\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To get this solution, work with each term individually. With each term, increase the exponent by \\(1\\) and divide by the new exponent. It is important to remember the integration constant \\(c\\), but this can be added at the very end.<\/p>\n<p style=\"text-align: center; line-height: 50px;\">\n\\(\\int x^4 \\, dx=\\frac{1}{5}x^5\\)<br \/>\n\\(\\int x^3 \\, dx=\\frac{1}{4}x^4\\)<br \/>\n\\(\\int x^2 \\, dx=\\frac{1}{3}x^3\\)<br \/>\n\\(\\int x \\, dx=\\frac{1}{2}x^2\\)<br \/>\n\\(\\int 1 \\, dx=x\\)<\/p>\n<p style=\"text-align: center; line-height: 45px;\">\\(\\int (x^4+x^3+x^2+x+1) \\, dx\\)\\(\\:=\\frac{1}{5}x^5+\\frac{1}{4}x^4+\\frac{1}{3}x^3+\\frac{1}{2}x^2+x+c\\)<\/p>\n<p>This solution can be checked by taking its derivative:<\/p>\n<p style=\"text-align:center; line-height: 45px;\">\\(\\frac{d}{dx}(\\frac{1}{5}x^5+\\frac{1}{4}x^4+\\frac{1}{3}x^3+\\frac{1}{2}x^2+x+c)\\)\\(\\:=x^4+x^3+x^2+x+1\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nEvaluate the indefinite integral below:<\/p>\n<div class=\"yellow-math-quote\">\\(\\int(-\\frac{3}{2}x^4+3x^2) \\, dx\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">\\(-\\frac{1}{5}x^5+3x^3+c\\)<\/div><div class=\"PQ\"  id=\"PQ-4-2\">\\(-3x^5+x^3+c\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-3\">\\(-\\frac{3}{10}x^5+x^3+c\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">\\(-6x^3+6x+c\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, notice that the integrand in this problem has multiple terms. Each term will be evaluated separately, beginning with the first one:<\/p>\n<p style=\"text-align: center\">\\(\\int -\\frac{3}{2}x^4 \\, dx\\)<\/p>\n<p>The fraction here can get in the way, so apply the property that constants can be factored out to move \\(-\\frac{3}{2}\\) outside the integral. The term is then \\(-\\frac{3}{2}\\int x^4 \\, dx\\), which is easier to solve. <\/p>\n<p>Increase the exponent by 1, then divide by that new exponent.<\/p>\n<p style=\"text-align: center;\">\\(-\\frac{3}{2}\\int x^4 \\, dx=-\\frac{3}{2}\\times\\frac{1}{5}x^5=-\\frac{3}{10}x^5\\)<\/p>\n<p>The second term of the integrand can be solved by again using the rule of increasing the exponent and dividing.<\/p>\n<p style=\"text-align:center;overflow-y:hidden;overflow-x:auto;line-height: 50px;\">\\(\\int 3x^2 \\, dx=\\frac{3}{2+1}x^{2+1}=x^3\\)<\/p>\n<p>The solution of the entire integral can be found by putting these two pieces together and adding the integration constant \\(c\\):<\/p>\n<p style=\"text-align:center;\">\\(\\int (-\\frac{3}{2}x^4+3x^2\u2009) \\, dx\\)\\(\\:=-\\frac{3}{10}x^5+x^3+c\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nCaleb went to a drag race recently to study the speed and displacement of the cars. He determined that the speed of one car during the race could be expressed as the function \\(f(x)=100x\\), where \\(x\\) represents time in seconds. Determine \\(F(x)\\), the function of the car\u2019s displacement over time \\(x\\), by taking the integral of \\(f(x)\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">\\(F(x)=50x^2+c\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\(F(x)=50x^2\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\(F(x)=100+c\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\">\\(F(x)=50x^2+100x+c\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Velocity and displacement have a fascinating relationship because displacement can be found by taking the integral of velocity. Conversely, velocity can be found by taking the derivative of displacement.<\/p>\n<p>In this problem, we find the function of displacement, \\(F(x)\\), by taking the integral of the function of velocity, \\(f(x)\\). In other words, simply evaluate the integral \\(\\int f(x)dx=\\int 100x\u2009dx\\).<\/p>\n<p>First, raise the power of \\(x\\) by 1, then divide by the new exponent. Finally, add the integration constant \\(c\\).<\/p>\n<p style=\"text-align:center;\">\\(\\int 100x\u2009\\, dx=\\frac{100}{1+1}x^{1+1}+c=50x^2+c\\)<\/p>\n<p>So \\(F(x)=50x^2+c\\) represents the position, or displacement, of the car during the race. This solution can be checked by taking its derivative:<\/p>\n<p style=\"text-align:center\">\\(\\frac{d}{dx}(50x^2+c)=100x\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/calculus\/\">Return to Calculus Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Calculus Videos<\/p>\n","protected":false},"author":1,"featured_media":156101,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-38229","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-calculus-videos","7":"page_category-long-math","8":"page_category-math-advertising-group","9":"page_category-video-pages-for-study-course-sidebar-ad","10":"page_type-video","11":"content_type-practice-questions","12":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.10 - aioseo.com -->\n\t<meta name=\"description\" content=\"An indefinite integral does not have any upper or lower limit. 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