{"id":12485,"date":"2018-10-01t13:18:43","date_gmt":"2018-10-01t20:18:43","guid":{"rendered":"\/\/www.catharsisit.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":"\/\/www.catharsisit.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=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-exam-review-integrals\/\">ap calculus exam review: integrals<\/a><\/li>\n<li><a href=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.com\/hs\/files\/2018\/04\/integral_lin_den.png 320w, \/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.com\/hs\/files\/2018\/04\/integral_arctan.png 359w, \/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.com\/hs\/files\/2018\/04\/trig_substitutions.png 547w, \/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.com\/hs\/files\/2018\/04\/trig_subs_parta.png 304w, \/\/www.catharsisit.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=\"\/\/www.catharsisit.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=\"\/\/www.catharsisit.com\/hs\/files\/2018\/04\/trig_subs_partb.png 486w, \/\/www.catharsisit.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=\"\/\/www.catharsisit.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],"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. in this short article, we&#039;ll take a look at some of the most common integrals on the test.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"common integrals on the ap calc exam\" \/>\n<meta property=\"og:description\" content=\"the ap calculus exams include a lot of integration. in this short article, we&#039;ll take a look at some of the most common integrals on the test.\" \/>\n<meta property=\"og:url\" content=\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/\" \/>\n<meta property=\"og:site_name\" content=\"magoosh blog | high school\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/magooshsat\/\" \/>\n<meta property=\"article:published_time\" content=\"2018-10-01t20:18:43+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2018-09-30t20:21:41+00:00\" \/>\n<meta property=\"og:image\" content=\"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/trig_antiderivatives.gif\" \/>\n<meta name=\"author\" content=\"shaun ault\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@shaunaultmath\" \/>\n<meta name=\"twitter:site\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:label1\" content=\"written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"shaun ault\" \/>\n\t<meta name=\"twitter:label2\" content=\"est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"article\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/#article\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/\"},\"author\":{\"name\":\"shaun ault\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\"},\"headline\":\"common integrals on the ap calc exam\",\"datepublished\":\"2018-10-01t20:18:43+00:00\",\"datemodified\":\"2018-09-30t20:21:41+00:00\",\"mainentityofpage\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/\"},\"wordcount\":547,\"commentcount\":0,\"publisher\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\"},\"keywords\":[\"ap calculus\"],\"articlesection\":[\"ap\"],\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"commentaction\",\"name\":\"comment\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/#respond\"]}]},{\"@type\":\"webpage\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/\",\"name\":\"common integrals on the ap calc exam - magoosh blog | high school\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#website\"},\"datepublished\":\"2018-10-01t20:18:43+00:00\",\"datemodified\":\"2018-09-30t20:21:41+00:00\",\"description\":\"the ap calculus exams include a lot of integration. in this short article, we'll take a look at some of the most common integrals on the test.\",\"breadcrumb\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/#breadcrumb\"},\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"readaction\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/\"]}]},{\"@type\":\"breadcrumblist\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/#breadcrumb\",\"itemlistelement\":[{\"@type\":\"listitem\",\"position\":1,\"name\":\"home\",\"item\":\"\/\/www.catharsisit.com\/hs\/\"},{\"@type\":\"listitem\",\"position\":2,\"name\":\"common integrals on the ap calc exam\"}]},{\"@type\":\"website\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#website\",\"url\":\"\/\/www.catharsisit.com\/hs\/\",\"name\":\"magoosh blog | high school\",\"description\":\"act, sat, college admissions, life\",\"publisher\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\"},\"potentialaction\":[{\"@type\":\"searchaction\",\"target\":{\"@type\":\"entrypoint\",\"urltemplate\":\"\/\/www.catharsisit.com\/hs\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inlanguage\":\"en-us\"},{\"@type\":\"organization\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\",\"name\":\"magoosh\",\"url\":\"\/\/www.catharsisit.com\/hs\/\",\"logo\":{\"@type\":\"imageobject\",\"inlanguage\":\"en-us\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png\",\"contenturl\":\"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png\",\"width\":265,\"height\":60,\"caption\":\"magoosh\"},\"image\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/\"},\"sameas\":[\"https:\/\/www.facebook.com\/magooshsat\/\",\"https:\/\/twitter.com\/magooshsat_act\"]},{\"@type\":\"person\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\",\"name\":\"shaun ault\",\"image\":{\"@type\":\"imageobject\",\"inlanguage\":\"en-us\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/7c80c2046678e19beb5b9c8401d56613\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g\",\"contenturl\":\"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g\",\"caption\":\"shaun ault\"},\"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!\",\"sameas\":[\"http:\/\/valdosta.academia.edu\/shaunault\",\"https:\/\/twitter.com\/shaunaultmath\"],\"url\":\"\/\/www.catharsisit.com\/hs\/author\/shaunault\/\"}]}<\/script>\n<!-- \/ yoast seo premium plugin. -->","yoast_head_json":{"title":"common integrals on the ap calc exam - magoosh blog | high school","description":"the ap calculus exams include a lot of integration. in this short article, we'll take a look at some of the most common integrals on the test.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/","og_locale":"en_us","og_type":"article","og_title":"common integrals on the ap calc exam","og_description":"the ap calculus exams include a lot of integration. in this short article, we'll take a look at some of the most common integrals on the test.","og_url":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/","og_site_name":"magoosh blog | high school","article_publisher":"https:\/\/www.facebook.com\/magooshsat\/","article_published_time":"2018-10-01t20:18:43+00:00","article_modified_time":"2018-09-30t20:21:41+00:00","og_image":[{"url":"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/trig_antiderivatives.gif"}],"author":"shaun ault","twitter_card":"summary_large_image","twitter_creator":"@shaunaultmath","twitter_site":"@magooshsat_act","twitter_misc":{"written by":"shaun ault","est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"article","@id":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/#article","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/"},"author":{"name":"shaun ault","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f"},"headline":"common integrals on the ap calc exam","datepublished":"2018-10-01t20:18:43+00:00","datemodified":"2018-09-30t20:21:41+00:00","mainentityofpage":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/"},"wordcount":547,"commentcount":0,"publisher":{"@id":"\/\/www.catharsisit.com\/hs\/#organization"},"keywords":["ap calculus"],"articlesection":["ap"],"inlanguage":"en-us","potentialaction":[{"@type":"commentaction","name":"comment","target":["\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/#respond"]}]},{"@type":"webpage","@id":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/","url":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/","name":"common integrals on the ap calc exam - magoosh blog | high school","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/#website"},"datepublished":"2018-10-01t20:18:43+00:00","datemodified":"2018-09-30t20:21:41+00:00","description":"the ap calculus exams include a lot of integration. in this short article, we'll take a look at some of the most common integrals on the test.","breadcrumb":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/#breadcrumb"},"inlanguage":"en-us","potentialaction":[{"@type":"readaction","target":["\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/"]}]},{"@type":"breadcrumblist","@id":"\/\/www.catharsisit.com\/hs\/ap\/common-integrals-on-the-ap-calc-exam\/#breadcrumb","itemlistelement":[{"@type":"listitem","position":1,"name":"home","item":"\/\/www.catharsisit.com\/hs\/"},{"@type":"listitem","position":2,"name":"common integrals on the ap calc exam"}]},{"@type":"website","@id":"\/\/www.catharsisit.com\/hs\/#website","url":"\/\/www.catharsisit.com\/hs\/","name":"magoosh blog | high school","description":"act, sat, college admissions, life","publisher":{"@id":"\/\/www.catharsisit.com\/hs\/#organization"},"potentialaction":[{"@type":"searchaction","target":{"@type":"entrypoint","urltemplate":"\/\/www.catharsisit.com\/hs\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inlanguage":"en-us"},{"@type":"organization","@id":"\/\/www.catharsisit.com\/hs\/#organization","name":"magoosh","url":"\/\/www.catharsisit.com\/hs\/","logo":{"@type":"imageobject","inlanguage":"en-us","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/","url":"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png","contenturl":"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png","width":265,"height":60,"caption":"magoosh"},"image":{"@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/"},"sameas":["https:\/\/www.facebook.com\/magooshsat\/","https:\/\/twitter.com\/magooshsat_act"]},{"@type":"person","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f","name":"shaun ault","image":{"@type":"imageobject","inlanguage":"en-us","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/7c80c2046678e19beb5b9c8401d56613","url":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","contenturl":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","caption":"shaun ault"},"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!","sameas":["http:\/\/valdosta.academia.edu\/shaunault","https:\/\/twitter.com\/shaunaultmath"],"url":"\/\/www.catharsisit.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\/28c36635b8fed6717752755f46cda239?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":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/12485"}],"collection":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/users\/223"}],"replies":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/comments?post=12485"}],"version-history":[{"count":0,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/12485\/revisions"}],"wp:attachment":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/media?parent=12485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/categories?post=12485"},{"taxonomy":"post_tag","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/tags?post=12485"},{"taxonomy":"author","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=12485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}