To find the antiderivatives of f(x)=sin\u2061(x)cos\u2061(x)f(x) = \\sin(x) \\cos(x)f(x)=sin(x)cos(x), we can simplify the expression using trigonometric identities. Here’s how you can do it:<\/p>\n\n\n\n
Step 1: Simplifying the Function<\/h3>\n\n\n\n
The function f(x)=sin\u2061(x)cos\u2061(x)f(x) = \\sin(x) \\cos(x)f(x)=sin(x)cos(x) can be rewritten using the double angle identity for sine:sin\u2061(2x)=2sin\u2061(x)cos\u2061(x)\\sin(2x) = 2\\sin(x)\\cos(x)sin(2x)=2sin(x)cos(x)<\/p>\n\n\n\n
Therefore, we can express sin\u2061(x)cos\u2061(x)\\sin(x)\\cos(x)sin(x)cos(x) as:sin\u2061(x)cos\u2061(x)=12sin\u2061(2x)\\sin(x) \\cos(x) = \\frac{1}{2} \\sin(2x)sin(x)cos(x)=21\u200bsin(2x)<\/p>\n\n\n\n
Now, the function becomes:f(x)=12sin\u2061(2x)f(x) = \\frac{1}{2} \\sin(2x)f(x)=21\u200bsin(2x)<\/p>\n\n\n\n
Step 2: Finding the Antiderivative<\/h3>\n\n\n\n
To find the antiderivative of 12sin\u2061(2x)\\frac{1}{2} \\sin(2x)21\u200bsin(2x), we can apply the standard antiderivative rule for sine functions, which is:\u222bsin\u2061(kx)\u2009dx=\u22121kcos\u2061(kx)\\int \\sin(kx) \\, dx = -\\frac{1}{k} \\cos(kx)\u222bsin(kx)dx=\u2212k1\u200bcos(kx)<\/p>\n\n\n\n
In this case, k=2k = 2k=2, so the antiderivative of sin\u2061(2x)\\sin(2x)sin(2x) is:\u221212cos\u2061(2x)-\\frac{1}{2} \\cos(2x)\u221221\u200bcos(2x)<\/p>\n\n\n\n
Thus, the antiderivative of 12sin\u2061(2x)\\frac{1}{2} \\sin(2x)21\u200bsin(2x) is:12\u22c5(\u221212cos\u2061(2x))=\u221214cos\u2061(2x)\\frac{1}{2} \\cdot \\left( -\\frac{1}{2} \\cos(2x) \\right) = -\\frac{1}{4} \\cos(2x)21\u200b\u22c5(\u221221\u200bcos(2x))=\u221241\u200bcos(2x)<\/p>\n\n\n\n
Step 3: Adding the Constant of Integration<\/h3>\n\n\n\n
We always add a constant CCC when finding an indefinite integral. So, the antiderivative is:F(x)=\u221214cos\u2061(2x)+CF(x) = -\\frac{1}{4} \\cos(2x) + CF(x)=\u221241\u200bcos(2x)+C<\/p>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n
The antiderivative of f(x)=sin\u2061(x)cos\u2061(x)f(x) = \\sin(x) \\cos(x)f(x)=sin(x)cos(x) is F(x)=\u221214cos\u2061(2x)+CF(x) = -\\frac{1}{4} \\cos(2x) + CF(x)=\u221241\u200bcos(2x)+C.<\/p>\n\n\n\n
\n\n\n\n
If there are multiple choices to pick from, check if this result is listed or closely matches one of the options.<\/p>\n\n\n\n Which of the following are anti-derivatives of f (x) = sinxcosx cos” cos(2x) [and II Il and MIL The Correct Answer and Explanation is: To find the antiderivatives of f(x)=sin\u2061(x)cos\u2061(x)f(x) = \\sin(x) \\cos(x)f(x)=sin(x)cos(x), we can simplify the expression using trigonometric identities. Here’s how you can do it: Step 1: Simplifying the Function The function f(x)=sin\u2061(x)cos\u2061(x)f(x) […]<\/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-244399","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244399","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=244399"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244399\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244399"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244399"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-447.jpeg\")
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