{"id":244395,"date":"2025-07-05T05:58:42","date_gmt":"2025-07-05T05:58:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=244395"},"modified":"2025-07-05T05:58:44","modified_gmt":"2025-07-05T05:58:44","slug":"consider-the-function-fx-sin-x-cos-x-select-one-sin-x-cos-x-is-an-antiderivative-of-fx","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/05\/consider-the-function-fx-sin-x-cos-x-select-one-sin-x-cos-x-is-an-antiderivative-of-fx\/","title":{"rendered":"Consider the function f(x) sin x + cos x Select one: sin x cos x is an antiderivative of f(x)"},"content":{"rendered":"\n

Consider the function f(x) sin x + cos x Select one: sin x cos x is an antiderivative of f(x)_ b. ~ sinx + cosx is an antiderivative of f(x) None of the other four answers is correct: sin X + cos X is an antiderivative of f(x) e. Sin x cos x is an antiderivative of f (x).<\/p>\n\n\n\n

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

The correct answer is: None of the other four answers is correct.<\/strong><\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

You are asked to determine which expression is an antiderivative of the function f(x)=sin\u2061(x)+cos\u2061(x)f(x) = \\sin(x) + \\cos(x)f(x)=sin(x)+cos(x).<\/p>\n\n\n\n

Antiderivative of f(x)f(x)f(x):<\/strong><\/p>\n\n\n\n

To find the antiderivative of f(x)=sin\u2061(x)+cos\u2061(x)f(x) = \\sin(x) + \\cos(x)f(x)=sin(x)+cos(x), we need to integrate the function with respect to xxx.\u222b(sin\u2061(x)+cos\u2061(x))\u2009dx\\int (\\sin(x) + \\cos(x)) \\, dx\u222b(sin(x)+cos(x))dx<\/p>\n\n\n\n

We can apply the basic rules of integration:<\/p>\n\n\n\n

    \n
  • The antiderivative of sin\u2061(x)\\sin(x)sin(x) is \u2212cos\u2061(x)-\\cos(x)\u2212cos(x), because ddx(\u2212cos\u2061(x))=sin\u2061(x)\\frac{d}{dx} (-\\cos(x)) = \\sin(x)dxd\u200b(\u2212cos(x))=sin(x).<\/li>\n\n\n\n
  • The antiderivative of cos\u2061(x)\\cos(x)cos(x) is sin\u2061(x)\\sin(x)sin(x), because ddx(sin\u2061(x))=cos\u2061(x)\\frac{d}{dx} (\\sin(x)) = \\cos(x)dxd\u200b(sin(x))=cos(x).<\/li>\n<\/ul>\n\n\n\n

    Thus, the antiderivative of f(x)=sin\u2061(x)+cos\u2061(x)f(x) = \\sin(x) + \\cos(x)f(x)=sin(x)+cos(x) is:\u2212cos\u2061(x)+sin\u2061(x)+C-\\cos(x) + \\sin(x) + C\u2212cos(x)+sin(x)+C<\/p>\n\n\n\n

    Where CCC is the constant of integration.<\/p>\n\n\n\n

    Review of the Options:<\/h3>\n\n\n\n
      \n
    1. sin(x) cos(x) is an antiderivative of f(x)f(x)f(x)<\/strong>: This is incorrect because the correct antiderivative does not involve a product of sine and cosine, but rather their individual integrals.<\/li>\n\n\n\n
    2. sin(x) + cos(x) is an antiderivative of f(x)f(x)f(x)<\/strong>: This is also incorrect because the derivative of sin\u2061(x)+cos\u2061(x)\\sin(x) + \\cos(x)sin(x)+cos(x) is cos\u2061(x)\u2212sin\u2061(x)\\cos(x) – \\sin(x)cos(x)\u2212sin(x), not sin\u2061(x)+cos\u2061(x)\\sin(x) + \\cos(x)sin(x)+cos(x).<\/li>\n\n\n\n
    3. None of the other four answers is correct<\/strong>: This is the correct choice because none of the other given options match the actual antiderivative.<\/li>\n\n\n\n
    4. sin(x) cos(x) is an antiderivative of f(x)f(x)f(x)<\/strong> (repeated): As explained above, this is incorrect.<\/li>\n<\/ol>\n\n\n\n

      Therefore, None of the other four answers is correct<\/strong> is the accurate answer. The correct antiderivative is \u2212cos\u2061(x)+sin\u2061(x)+C-\\cos(x) + \\sin(x) + C\u2212cos(x)+sin(x)+C.<\/p>\n\n\n\n

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

      Consider the function f(x) sin x + cos x Select one: sin x cos x is an antiderivative of f(x)_ b. ~ sinx + cosx is an antiderivative of f(x) None of the other four answers is correct: sin X + cos X is an antiderivative of f(x) e. Sin x cos x is an […]<\/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-244395","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244395","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=244395"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244395\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244395"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244395"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244395"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}