Evaluate integral sinx\/x dx as an infinite series.<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n To evaluate the integral ( \\int \\frac{\\sin x}{x} \\, dx ) as an infinite series, we can start by expressing ( \\frac{\\sin x}{x} ) as a Taylor series expansion and then integrate term by term.<\/p>\n\n\n\n The Taylor series expansion of ( \\sin x ) around ( x = 0 ) is given by:<\/p>\n\n\n\n [ Now, divide the series for ( \\sin x ) by ( x ) to obtain ( \\frac{\\sin x}{x} ):<\/p>\n\n\n\n [ Now, we can integrate ( \\frac{\\sin x}{x} ) term by term:<\/p>\n\n\n\n [ Integrating each term individually:<\/p>\n\n\n\n [ Thus, the integral becomes:<\/p>\n\n\n\n [ where ( C ) is the constant of integration.<\/p>\n\n\n\n [ Evaluate integral sinx\/x dx as an infinite series. The correct answer and explanation is : To evaluate the integral ( \\int \\frac{\\sin x}{x} \\, dx ) as an infinite series, we can start by expressing ( \\frac{\\sin x}{x} ) as a Taylor series expansion and then integrate term by term. Step 1: Taylor Series for […]<\/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-209707","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209707","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=209707"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209707\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=209707"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=209707"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=209707"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Taylor Series for ( \\sin x )<\/h3>\n\n\n\n
\\sin x = x – \\frac{x^3}{3!} + \\frac{x^5}{5!} – \\frac{x^7}{7!} + \\cdots
]<\/p>\n\n\n\nStep 2: Divide by ( x )<\/h3>\n\n\n\n
\\frac{\\sin x}{x} = 1 – \\frac{x^2}{3!} + \\frac{x^4}{5!} – \\frac{x^6}{7!} + \\cdots
]<\/p>\n\n\n\nStep 3: Integrate Term by Term<\/h3>\n\n\n\n
\\int \\frac{\\sin x}{x} \\, dx = \\int \\left( 1 – \\frac{x^2}{3!} + \\frac{x^4}{5!} – \\frac{x^6}{7!} + \\cdots \\right) dx
]<\/p>\n\n\n\n
\\int 1 \\, dx = x
]
[
\\int \\frac{x^2}{3!} \\, dx = \\frac{x^3}{3! \\cdot 3} = \\frac{x^3}{18}
]
[
\\int \\frac{x^4}{5!} \\, dx = \\frac{x^5}{5! \\cdot 5} = \\frac{x^5}{600}
]
[
\\int \\frac{x^6}{7!} \\, dx = \\frac{x^7}{7! \\cdot 7} = \\frac{x^7}{35280}
]
And so on.<\/p>\n\n\n\n
\\int \\frac{\\sin x}{x} \\, dx = x – \\frac{x^3}{18} + \\frac{x^5}{600} – \\frac{x^7}{35280} + \\cdots + C
]<\/p>\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
\\int \\frac{\\sin x}{x} \\, dx = x – \\frac{x^3}{18} + \\frac{x^5}{600} – \\frac{x^7}{35280} + \\cdots + C
]<\/p>\n","protected":false},"excerpt":{"rendered":"