{"id":8734,"date":"2017-01-27t09:28:00","date_gmt":"2017-01-27t17:28:00","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=8734"},"modified":"2017-01-25t20:28:30","modified_gmt":"2017-01-26t04:28:30","slug":"ap-calculus-review-indefinite-integrals","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/","title":{"rendered":"ap calculus review: indefinite integrals"},"content":{"rendered":"

indefinite integrals make up a substantial part of what is covered on the ap calculus ab and bc exams. in this review article, we highlight a few concepts and techniques that you’ll need to be familiar with.<\/p>\n

what are indefinite integrals?<\/h2>\n

there are two kinds of integrals, the definite and indefinite integrals. this article only discusses indefinite integrals. for a more general overview, including information about definite integrals, check out this review of integrals<\/a>.<\/p>\n

an indefinite integral<\/strong> of a function f<\/em> is the most general antiderivative<\/strong> of f<\/em>.<\/p>\n

\"indefinite<\/p>\n

here, the function f<\/em> is any particular antiderivative for f<\/em>. that is, f<\/em>\u00a0‘<\/sup>(x<\/em>) = f<\/em>(x<\/em>). for example, f<\/em>(x<\/em>) = x<\/em>2<\/sup> is an antiderivative for f<\/em>(x<\/em>) = 2x<\/em>, since (x<\/em>2<\/sup>)’ = 2x<\/em>.<\/p>\n

the c<\/em> is the constant of integration<\/strong>. it stands for any constant, and it must be part of your answer to an indefinite integral.<\/p>\n

so for example,<\/p>\n

\"integral<\/p>\n

what’s the deal with the “\u00a0+\u00a0c<\/em>\u00a0” anyway?<\/h3>\n

the reason we need to tack on that “\u00a0+\u00a0c<\/em>\u00a0” is so that we can describe absolutely every antiderivative for f<\/em>. remember the derivative rule for constant functions:<\/p>\n

\"the<\/p>\n

therefore, if there is a particular function f<\/em>(x<\/em>) such that f<\/em>\u00a0‘(x<\/em>) = f<\/em>(x<\/em>), then for any constant c<\/em>, we have:<\/p>\n

\"derivative<\/p>\n

thus the most general<\/em> antiderivative of f<\/em>(x<\/em>) would be f<\/em>(x<\/em>) + c<\/em>.<\/p>\n

indefinite integral techniques<\/h2>\n

most everyone knows that you shouldn’t use a screwdriver to pound in a nail. and hammers do not help when driving in screws. in a similar way, you should be aware that each indefinite integral problem requires its own set of tools.<\/p>\n

we’ll discuss a few integration tools, including the basic antiderivative rules, substitution, integration by parts, and partial fractions. other more advanced tools may be covered in future magoosh articles.<\/p>\n

also, it’s important to realize that each technique requires quite a bit of practice before you can really get good at it. don’t expect to become an expert on the first day.<\/p>\n

basic antiderivative rules<\/h3>\n

these rules are really just derivative rules<\/a> in reverse. here is a list of the basic antiderivative rules.<\/p>\n

\"power<\/p>\n

\"sum<\/p>\n

\"constant<\/p>\n

\"constant<\/p>\n

\"rule<\/p>\n

\"exponential<\/p>\n

\"trigonometric<\/p>\n

substitution<\/h3>\n

the substitution rule<\/strong>, or as it’s more commonly known, u<\/em>-substitution, is a rule that “reverses” the chain rule.<\/p>\n

\"substitution<\/p>\n

this rule helps when the integrand is a composition of two functions. that is, if there is a function inside<\/em> another function. for example, we would identify (5x<\/em> + 1)8<\/sup> as a composition of the functions u<\/em> = 5x<\/em> + 1 and f<\/em>(u<\/em>) = u<\/em>8<\/sup>. so if we needed to know the indefinite integral of (5x<\/em> + 1)8<\/sup>, we could use substitution.<\/p>\n

steps for substitution<\/h4>\n

substitution can be difficult because the formula requires a specific setup. however, if you follow the steps outlined below, then you’ll be sure to get it right every time.<\/p>\n

    \n
  1. identify a part of the function that you will try to substitute, and write it down: u<\/em> = g<\/em>(x<\/em>). it may not be obvious what to pick, so don’t be afraid of a little trial and error at first.<\/li>\n
  2. take the differential<\/strong> of your substitution. that is, find the derivative of g<\/em> and write it in the form, du<\/em> = g<\/em>\u00a0‘(x<\/em>)\u00a0dx<\/em>.<\/li>\n
  3. substitute both u<\/em> and du<\/em> into the original integral. this may involve solving the differential for dx<\/em> and then replacing the dx<\/em> in the integral.<\/li>\n
  4. if the new integral involves only u<\/em> and du<\/em>, then simplify and integrate using standard methods.<\/li>\n
  5. finally, plug u<\/em> = g<\/em>(x<\/em>) back in so that your answer is in terms of the original variable x<\/em>.<\/li>\n<\/ol>\n

    using the substitution rule<\/h4>\n

    \"substitution<\/p>\n

    first we must decide what to substitute. experience tells us to look for expressions within parentheses.<\/p>\n

    don’t forget to take the differential. i find it helpful to solve for dx<\/em>.<\/p>\n

    \"substitution<\/p>\n

    now you can replace 5x<\/em> + 1 by u<\/em> and dx<\/em> by (1\/5)du<\/em>. then integrate and finally plug back in u = 5x<\/em> + 1<\/em>.<\/p>\n

    \"substitution<\/p>\n

    integration by parts<\/h3>\n

    integration by parts<\/strong> (ibp) is a powerful method that may be used when there are certain kinds of products in the integrand. in fact, you can think of ibp as a way to “reverse” the product rule.<\/p>\n

    suppose u<\/em> and v<\/em> are differentiable functions of x<\/em>. then the ibp formula states that:<\/p>\n

    \"integration<\/p>\n

    example using ibp<\/h4>\n

    typically we use ibp when there are products of powers of x<\/em>, exponential functions, and\/or trigonometric functions in the integrand.<\/p>\n

    \"ibp<\/p>\n

    here we will choose u<\/em> = 3x<\/em>, and dv<\/em> = cos x<\/em> dx<\/em>. again, experience guides our choices. if you had chosen the functions the other way around, then the integral would have gotten more complicated.<\/p>\n

    now find du<\/em> by taking a derivative, and v<\/em> by integrating.<\/p>\n

    \"ibp,<\/p>\n

    next, we use the ibp formula to rewrite the original integral in a different way. try to track where u<\/em>, v<\/em>, du<\/em>, and dv<\/em> show up in the problem. (hint:<\/em> they’re color coded.)<\/p>\n

    \"ibp<\/p>\n

    partial fractions<\/h3>\n

    last but not least, let’s talk about the method of partial fractions<\/strong> (pf). we can use pf whenever the integrand is a rational function<\/em> whose denominator has a\u00a0degree\u00a0of at least 2. the main idea is to break apart the fraction into a sum of\u00a0simpler fractions.<\/p>\n

    in this short review, there is not enough time to explain all of the details. so if you’re interested in learning more, check out this article<\/a>.<\/p>\n

    instead, let’s see a quick example of the technique in action.<\/p>\n

    \"partial<\/p>\n

    the key is to use your algebra skills to factor the denominator and split into two fractions, solving for the unknown constants in each numerator.<\/p>\n

    \"partial<\/p>\n

    it can be determined that a<\/em> = -2 and b<\/em> = 3 in this example. again, because this article is just review, we leave some of the details to you.<\/p>\n

    now we can work out the problem completely.<\/p>\n

    \"example<\/p>\n

    summary<\/h2>\n

    indefinite integral problems come in many different types on the ap calculus exams. remember that an indefinite integral is the most general antiderivative of a function.<\/p>\n

    among the wide range of techniques available, most problems can be handled by one or more of the following methods.<\/p>\n

      \n
    • basic antiderivative formulas, including the power rule and rules for special kinds of functions (such as trigonometric and exponential).<\/li>\n
    • substitution<\/li>\n
    • integration by parts<\/li>\n
    • partial fractions<\/li>\n<\/ul>\n

      after much practice, you will be able to choose the best technique for each integral problem. just like a good carpenter, using the right tool makes the job easy. you may even come to enjoy the challenge of indefinite integrals!<\/p>\n","protected":false},"excerpt":{"rendered":"

      indefinite integrals make up a substantial part of what is covered on the ap calculus ab and bc exams. in this review article, we highlight a few concepts and techniques that you’ll need to be familiar with. what are indefinite integrals? there are two kinds of integrals, the definite and indefinite integrals. this article only […]<\/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-8734","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"acf":[],"yoast_head":"\nap calculus review: indefinite integrals - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"indefinite integrals make up a substantial part of the ap calculus ab and bc exams. click here to learn the concepts and techniques that you'll need.\" \/>\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\/ap-calculus-review-indefinite-integrals\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"ap calculus review: indefinite integrals\" \/>\n<meta property=\"og:description\" content=\"indefinite integrals make up a substantial part of the ap calculus ab and bc exams. click here to learn the concepts and techniques that you'll need.\" \/>\n<meta property=\"og:url\" content=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/\" \/>\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=\"2017-01-27t17:28:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2017-01-26t04:28:30+00:00\" \/>\n<meta property=\"og:image\" content=\"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/indefinite_integral.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=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"article\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/#article\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/\"},\"author\":{\"name\":\"shaun ault\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\"},\"headline\":\"ap calculus review: indefinite integrals\",\"datepublished\":\"2017-01-27t17:28:00+00:00\",\"datemodified\":\"2017-01-26t04:28:30+00:00\",\"mainentityofpage\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/\"},\"wordcount\":1058,\"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\/ap-calculus-review-indefinite-integrals\/#respond\"]}]},{\"@type\":\"webpage\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/\",\"name\":\"ap calculus review: indefinite integrals - magoosh blog | high school\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#website\"},\"datepublished\":\"2017-01-27t17:28:00+00:00\",\"datemodified\":\"2017-01-26t04:28:30+00:00\",\"description\":\"indefinite integrals make up a substantial part of the ap calculus ab and bc exams. click here to learn the concepts and techniques that you'll need.\",\"breadcrumb\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/#breadcrumb\"},\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"readaction\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/\"]}]},{\"@type\":\"breadcrumblist\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/#breadcrumb\",\"itemlistelement\":[{\"@type\":\"listitem\",\"position\":1,\"name\":\"home\",\"item\":\"\/\/www.catharsisit.com\/hs\/\"},{\"@type\":\"listitem\",\"position\":2,\"name\":\"ap calculus review: indefinite integrals\"}]},{\"@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":"ap calculus review: indefinite integrals - magoosh blog | high school","description":"indefinite integrals make up a substantial part of the ap calculus ab and bc exams. click here to learn the concepts and techniques that you'll need.","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\/ap-calculus-review-indefinite-integrals\/","og_locale":"en_us","og_type":"article","og_title":"ap calculus review: indefinite integrals","og_description":"indefinite integrals make up a substantial part of the ap calculus ab and bc exams. click here to learn the concepts and techniques that you'll need.","og_url":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/","og_site_name":"magoosh blog | high school","article_publisher":"https:\/\/www.facebook.com\/magooshsat\/","article_published_time":"2017-01-27t17:28:00+00:00","article_modified_time":"2017-01-26t04:28:30+00:00","og_image":[{"url":"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/indefinite_integral.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":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"article","@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/#article","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/"},"author":{"name":"shaun ault","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f"},"headline":"ap calculus review: indefinite integrals","datepublished":"2017-01-27t17:28:00+00:00","datemodified":"2017-01-26t04:28:30+00:00","mainentityofpage":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/"},"wordcount":1058,"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\/ap-calculus-review-indefinite-integrals\/#respond"]}]},{"@type":"webpage","@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/","url":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/","name":"ap calculus review: indefinite integrals - magoosh blog | high school","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/#website"},"datepublished":"2017-01-27t17:28:00+00:00","datemodified":"2017-01-26t04:28:30+00:00","description":"indefinite integrals make up a substantial part of the ap calculus ab and bc exams. click here to learn the concepts and techniques that you'll need.","breadcrumb":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/#breadcrumb"},"inlanguage":"en-us","potentialaction":[{"@type":"readaction","target":["\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/"]}]},{"@type":"breadcrumblist","@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-indefinite-integrals\/#breadcrumb","itemlistelement":[{"@type":"listitem","position":1,"name":"home","item":"\/\/www.catharsisit.com\/hs\/"},{"@type":"listitem","position":2,"name":"ap calculus review: indefinite integrals"}]},{"@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\/8734"}],"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=8734"}],"version-history":[{"count":0,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/8734\/revisions"}],"wp:attachment":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/media?parent=8734"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/categories?post=8734"},{"taxonomy":"post_tag","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/tags?post=8734"},{"taxonomy":"author","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=8734"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}