{"id":12485,"date":"2018-10-01T13:18:43","date_gmt":"2018-10-01T20:18:43","guid":{"rendered":"https:\/\/magoosh.com\/hs\/?p=12485"},"modified":"2018-09-30T13:21:41","modified_gmt":"2018-09-30T20:21:41","slug":"common-integrals-on-the-ap-calc-exam","status":"publish","type":"post","link":"https:\/\/magoosh.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/","title":{"rendered":"Common Integrals on the AP Calc Exam"},"content":{"rendered":"<p>The AP Calculus exams include a substantial amount of integration.  So it&#8217;s very important to be familiar with integrals, numerous integration methods, and the interpretations and applications of integration.  In this short article, we&#8217;ll take a look at some of the most common integrals on the test.<\/p>\n<p>For a quick review of <em>integration<\/em> (or, <em>antidifferentiation<\/em>), you might want to check out the following articles first.<\/p>\n<ul>\n<li><a href=\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-exam-review-integrals\/\">AP Calculus Exam Review: Integrals<\/a><\/li>\n<li><a href=\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-exam-review-antidifferentiation\/\">AP Calculus Exam Review: Antidifferentiation<\/a><\/li>\n<\/ul>\n<p>And now, without further ado, here are some of the most common integrals found on the AP Calculus exams!<\/p>\n<h2>Common Integrals<\/h2>\n<p>The following seven integrals (or their close cousins) seem to pop up all the time on the AP Calculus AB and BC exams.<\/p>\n<h3>1. Remember your Trig Integrals!<\/h3>\n<p>Trigonometric functions are popular on the exam!  <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2017\/01\/Trig_Antiderivatives.gif\" alt=\"Trigonometric Antiderivatives\" width=\"455\" height=\"274\" class=\"aligncenter size-full wp-image-8742\" \/><\/p>\n<h3>2. Simple Substitutions<\/h3>\n<p>You need to recognize when to use the substitution <em>u<\/em> = <em>kx<\/em>, for constant <em>k<\/em>.  This substitution generates a factor of 1\/<em>k<\/em> because <em>du<\/em> = <em>k<\/em> <em>dx<\/em>.<\/p>\n<p>For example, <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/03\/integral_exp6x.png\" alt=\"integral of exp(6x)\" width=\"163\" height=\"41\" class=\"aligncenter size-full wp-image-12487\" \/><\/p>\n<h3>3. Common Integration By Parts<\/h3>\n<p>Integrands of the form <em>x<\/em> <em>f<\/em>(<em>x<\/em>) often lend themselves to <a href=\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-bc-review-integration-parts\/\">integration by parts (IBP)<\/a>.<\/p>\n<p>In the following integral, let <em>u<\/em> = <em>x<\/em> and <em>dv<\/em> = sin <em>x<\/em> <em>dx<\/em>, and use IBP.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/03\/integral_xsinx.png\" alt=\"integral of x sin x\" width=\"275\" height=\"41\" class=\"aligncenter size-full wp-image-12488\" \/><\/p>\n<h3>4. Linear Denominators<\/h3>\n<p>Integrands of the form <em>a<\/em>\/(<em>bx<\/em> + <em>c<\/em>) pop up as a result of partial fractions decomposition.  (See <a href=\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-bc-review-partial-fractions\/\">AP Calculus BC Review: Partial Fractions<\/a>).  While partial fractions is a BC test topic, it&#8217;s not rare to see an integral with linear denominator showing up in the AB test as well.<\/p>\n<p>The key is that substituting <em>u<\/em> = <em>bx<\/em> + <em>c<\/em> (and <em>du<\/em> = <em>b<\/em> <em>dx<\/em>) turns the integrand into a constant times 1\/<em>u<\/em>.  Let&#8217;s see how this works in general.  Keep in mind that <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> must be constants in order to use this rule.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/integral_lin_den.png\" alt=\"integral of a\/(bx+c)\" width=\"320\" height=\"79\" class=\"aligncenter size-full wp-image-12497\" srcset=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/integral_lin_den.png 320w, https:\/\/magoosh.com\/hs\/files\/2018\/04\/integral_lin_den-300x74.png 300w\" sizes=\"(max-width: 320px) 100vw, 320px\" \/><\/p>\n<h3>5. Integral of Ln <em>x<\/em><\/h3>\n<p>The antiderivative of <em>f<\/em>(<em>x<\/em>) = ln <em>x<\/em> is interesting.  You have to use a tricky integration by parts.<\/p>\n<p>Let <em>u<\/em> = ln <em>x<\/em>, and <em>dv<\/em> = <em>dx<\/em>.  <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/integal_lnx.png\" alt=\"integral of ln x\" width=\"243\" height=\"111\" class=\"aligncenter size-full wp-image-12498\" \/><\/p>\n<p>By the way, this trick works for other inverse functions too, such as the inverse trig functions, arcsin <em>x<\/em>, arccos <em>x<\/em>, and arctan <em>x<\/em>.  For example,<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/integral_arctan.png\" alt=\"integral of arctan x\" width=\"359\" height=\"83\" class=\"aligncenter size-full wp-image-12499\" srcset=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/integral_arctan.png 359w, https:\/\/magoosh.com\/hs\/files\/2018\/04\/integral_arctan-300x69.png 300w\" sizes=\"(max-width: 359px) 100vw, 359px\" \/><\/p>\n<h3>6. Using Trig Identities<\/h3>\n<p>For some trigonometric integrals, you have to rewrite the integrand in an equivalent way.  In other words, use a trig identity before integrating.  One of the most popular (and useful) techniques is the half-angle identity.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/integral_sin_sq.png\" alt=\"common integrals of sin^2 x\" width=\"278\" height=\"134\" class=\"aligncenter size-full wp-image-12500\" \/><\/p>\n<h3>7. Trigonometric Substitution<\/h3>\n<p>It&#8217;s no secret that the AP Calculus exams consist of challenging problems.  Perhaps the most challenging integrals are those that require a trigonometric substitution.<\/p>\n<p>The table below summarizes the trigonometric substitutions.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_substitutions.png\" alt=\"Table of trig substitutions\" width=\"547\" height=\"92\" class=\"aligncenter size-full wp-image-12501\" srcset=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_substitutions.png 547w, https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_substitutions-300x50.png 300w\" sizes=\"(max-width: 547px) 100vw, 547px\" \/><\/p>\n<p>For example, find the integral:  <img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_subs_problem.png\" alt=\"trig integral problem\" width=\"96\" height=\"45\" class=\"alignnone size-full wp-image-12503\" \/><\/p>\n<p>Here, the best substitution would be <em>x<\/em> = (3\/2) sin &theta;.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_subs_partA.png\" alt=\"trig_subs_partA\" width=\"304\" height=\"144\" class=\"aligncenter size-full wp-image-12502\" srcset=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_subs_partA.png 304w, https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_subs_partA-300x142.png 300w\" sizes=\"(max-width: 304px) 100vw, 304px\" \/><\/p>\n<p>Now we&#8217;re not out of the woods yet.  Use the half-angle identity (see point 6 above).  We also get to use the double-angle identity for sine in the second line.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_subs_partB.png\" alt=\"final part of solution for example trig substitution problem\" width=\"486\" height=\"193\" class=\"aligncenter size-full wp-image-12504\" srcset=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_subs_partB.png 486w, https:\/\/magoosh.com\/hs\/files\/2018\/04\/trig_subs_partB-300x119.png 300w\" sizes=\"(max-width: 486px) 100vw, 486px\" \/><\/p>\n<p>Note, the third line may seem like it comes out of nowhere.  But it&#8217;s based on the substitution and a right triangle.<\/p>\n<p>If <em>x<\/em> = (3\/2) sin &theta;, then sin &theta; = (2<em>x<\/em>) \/ 3.  Draw a right triangle with angle &theta;, opposite side 2<em>x<\/em>, and hypotenuse 3.<\/p>\n<p>By the Pythagorean Theorem, we find the adjacent side is equal to: <img decoding=\"async\" src=\"https:\/\/magoosh.com\/hs\/files\/2018\/04\/sqrt_9-4xsq.png\" alt=\"square root of 9 - 4x^2\" width=\"72\" height=\"19\" class=\"alignnone size-full wp-image-12505\" \/><\/p>\n<p>That allows us to identify cos &theta; in the expression (adjacent over hypotenuse).<\/p>\n<p>Finally, &theta; by itself is equal to arcsin(2<em>x<\/em>\/3).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The AP Calculus exams include a substantial amount of integration. So it&#8217;s very important to be familiar with integrals, numerous integration methods, and the interpretations and applications of integration. In this short article, we&#8217;ll take a look at some of the most common integrals on the test. For a quick review of integration (or, antidifferentiation), [&hellip;]<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"class_list":["post-12485","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Common Integrals on the AP Calc Exam - Magoosh Blog | High School<\/title>\n<meta name=\"description\" content=\"The AP Calculus exams include a lot of integration. 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D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!","sameAs":["http:\/\/valdosta.academia.edu\/ShaunAult","https:\/\/twitter.com\/ShaunAultMath"],"url":"https:\/\/magoosh.com\/hs\/author\/shaunault\/"}]}},"authors":[{"term_id":24932,"user_id":223,"is_guest":0,"slug":"shaunault","display_name":"Shaun Ault","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/f10cdb687137bc0ad4e885404588101b7cd4aa01ae2be48abda61f14fa3715e2?s=96&d=mm&r=g","user_url":"http:\/\/valdosta.academia.edu\/ShaunAult","last_name":"Ault","first_name":"Shaun","description":"Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition.  Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass.  Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!"}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/posts\/12485","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/users\/223"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/comments?post=12485"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/posts\/12485\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/media?parent=12485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/categories?post=12485"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/tags?post=12485"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=12485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}