{"id":259679,"date":"2025-07-19T05:06:06","date_gmt":"2025-07-19T05:06:06","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=259679"},"modified":"2025-07-19T05:06:09","modified_gmt":"2025-07-19T05:06:09","slug":"%e2%88%ab-x%c2%b2-e%e2%81%bb%c2%b2%cb%a3-dx","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/19\/%e2%88%ab-x%c2%b2-e%e2%81%bb%c2%b2%cb%a3-dx\/","title":{"rendered":"\u00a0\u222b x\u00b2 e\u207b\u00b2\u02e3 dx"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Here is the solution and explanation for problem number 11 from the image.<\/p>\n\n\n\n

Problem 11:<\/strong> \u222b x\u00b2 e\u207b\u00b2\u02e3 dx<\/p>\n\n\n\n

Correct Answer:<\/strong> – (1\/2)x\u00b2e\u207b\u00b2\u02e3 – (1\/2)xe\u207b\u00b2\u02e3 – (1\/4)e\u207b\u00b2\u02e3 + C<\/p>\n\n\n\n

Explanation:<\/strong><\/p>\n\n\n\n

This integral is best solved using the technique of integration by parts, which is applied when the integrand is a product of two different types of functions. The formula for integration by parts is \u222b u dv = uv – \u222b v du. The key to this method is choosing the parts u and dv correctly. A common guideline is the LIPET or LIATE rule (Logarithmic, Inverse Trig, Polynomial\/Algebraic, Exponential\/Trigonometric), which suggests the order for choosing u. In this case, we have a polynomial term (x\u00b2) and an exponential term (e\u207b\u00b2\u02e3), so we choose the polynomial as u.<\/p>\n\n\n\n

For the first application of integration by parts:
Let u = x\u00b2 and dv = e\u207b\u00b2\u02e3 dx.
Then we find du by differentiating u, and v by integrating dv:
du = 2x dx
v = \u222b e\u207b\u00b2\u02e3 dx = -1\/2 e\u207b\u00b2\u02e3<\/p>\n\n\n\n

Plugging these into the formula gives:
\u222b x\u00b2 e\u207b\u00b2\u02e3 dx = (x\u00b2)(-1\/2 e\u207b\u00b2\u02e3) – \u222b (-1\/2 e\u207b\u00b2\u02e3)(2x dx)
= -1\/2 x\u00b2 e\u207b\u00b2\u02e3 + \u222b x e\u207b\u00b2\u02e3 dx<\/p>\n\n\n\n

The new integral, \u222b x e\u207b\u00b2\u02e3 dx, still requires integration by parts. We apply the process again:
Let u = x and dv = e\u207b\u00b2\u02e3 dx.
Then du = dx and v = -1\/2 e\u207b\u00b2\u02e3.<\/p>\n\n\n\n

Applying the formula to this second integral:
\u222b x e\u207b\u00b2\u02e3 dx = (x)(-1\/2 e\u207b\u00b2\u02e3) – \u222b (-1\/2 e\u207b\u00b2\u02e3) dx
= -1\/2 xe\u207b\u00b2\u02e3 + 1\/2 \u222b e\u207b\u00b2\u02e3 dx
= -1\/2 xe\u207b\u00b2\u02e3 + 1\/2 (-1\/2 e\u207b\u00b2\u02e3)
= -1\/2 xe\u207b\u00b2\u02e3 – 1\/4 e\u207b\u00b2\u02e3<\/p>\n\n\n\n

Now, we substitute this result back into our first equation:
\u222b x\u00b2 e\u207b\u00b2\u02e3 dx = -1\/2 x\u00b2 e\u207b\u00b2\u02e3 + (-1\/2 xe\u207b\u00b2\u02e3 – 1\/4 e\u207b\u00b2\u02e3)<\/p>\n\n\n\n

Finally, we combine the terms and add the constant of integration, C, to get the final answer:
-1\/2 x\u00b2e\u207b\u00b2\u02e3 – 1\/2 xe\u207b\u00b2\u02e3 – 1\/4 e\u207b\u00b2\u02e3 + C<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

The Correct Answer and Explanation is: Here is the solution and explanation for problem number 11 from the image. Problem 11: \u222b x\u00b2 e\u207b\u00b2\u02e3 dx Correct Answer: – (1\/2)x\u00b2e\u207b\u00b2\u02e3 – (1\/2)xe\u207b\u00b2\u02e3 – (1\/4)e\u207b\u00b2\u02e3 + C Explanation: This integral is best solved using the technique of integration by parts, which is applied when the integrand is a product […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-259679","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259679","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=259679"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259679\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=259679"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=259679"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=259679"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}