{"id":567,"date":"2019-10-29T13:11:51","date_gmt":"2019-10-29T13:11:51","guid":{"rendered":"http:\/\/adrianbell.me\/?p=567"},"modified":"2019-10-29T13:18:39","modified_gmt":"2019-10-29T13:18:39","slug":"integration-revision-integration-by-substitution","status":"publish","type":"post","link":"http:\/\/adrianbell.me\/?p=567","title":{"rendered":"Integration Revision - Integration By Substitution"},"content":{"rendered":"

Now, looking at the other integral from Oct 13th...<\/p>\n

\"\int^{x}_{0}4xe^{-2x^{2}}\:dx=1-e^{-2x^{2}}\"<\/p>\n

In my opinion, this was scarier than the first, but looking at it again, what makes it scary is the various powers. e to the power of x to the power of 2 and such.<\/p>\n

But why should this be scary? There's practically only one tool that one could use to solve this: the only tool that includes functions of functions, integration by substitution:<\/p>\n

\"\int f(g(x))\:g^{\prime}(x)\:dx=\int f(u)\: du\"<\/p>\n

In our case, the inner function:
\n\"g(x)=-2x^{2}=u\"
\nso:
\n\"g^{\prime}(x)=-4x\"<\/p>\n

and the outer function:
\n\"f(x)=e^{x}\"<\/p>\n

Plugging all these into our \"integration by substitution\" tool gives:<\/p>\n

\"\int -4xe^{-2x^{2}}\:dx\"<\/p>\n

But this is slightly different to the original integral due to that minus sign. Of course this is trivial to deal with as:<\/p>\n

\"\int^{x}_{0}4xe^{-2x^{2}}\:dx=-\int^{x}_{0} -4xe^{-2x^{2}}\:dx\"<\/p>\n

Great! So now we have the structure we require to apply integration by substitution! We can substitute all of that with \"f(u)\", and all the scary bits go away. So:<\/p>\n

\"\int^{x}_{0}4xe^{-2x^{2}}\:dx\"<\/p>\n

\"=-\int^{x}_{0} -4xe^{-2x^{2}}\:dx\"<\/p>\n

\"=-\int^{u}_{0} e^{u}\:du\"<\/p>\n

\"=-[ e^{u}]^{u}_{0}\"<\/p>\n

\"=-(e^{u}-1)\"<\/p>\n

\"=1-e^{u}\"<\/p>\n

Then substituting \"u\" (=\"g(x)\") back in:<\/p>\n

\"=1-e^{-2x^{2}}\"<\/p>\n","protected":false},"excerpt":{"rendered":"

Now, looking at the other integral from Oct 13th... In my opinion, this was scarier than the first, but looking at it again, what makes it scary is the various powers. e to the power of x to the power of 2 and such. But why should this be scary? There's practically only one tool … Continue reading Integration Revision - Integration By Substitution<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[38],"tags":[],"class_list":["post-567","post","type-post","status-publish","format-standard","hentry","category-calculus"],"_links":{"self":[{"href":"http:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/posts\/567","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/adrianbell.me\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=567"}],"version-history":[{"count":10,"href":"http:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/posts\/567\/revisions"}],"predecessor-version":[{"id":577,"href":"http:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/posts\/567\/revisions\/577"}],"wp:attachment":[{"href":"http:\/\/adrianbell.me\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=567"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/adrianbell.me\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=567"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/adrianbell.me\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=567"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}