{"id":233203,"date":"2025-06-12T20:15:44","date_gmt":"2025-06-12T20:15:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233203"},"modified":"2025-06-12T20:17:47","modified_gmt":"2025-06-12T20:17:47","slug":"points-each-part-evaluate-the-integral-sin-2t-d","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/12\/points-each-part-evaluate-the-integral-sin-2t-d\/","title":{"rendered":"Evaluate the integral “sin 2t d"},"content":{"rendered":"\n

Evaluate the integral “sin 2t d. (). Determine the convolution flt) = e2 COS What is the corresponding F(8)?’<\/p>\n\n\n\n

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

Problem 1: Evaluate the integral of sin(2t)<\/strong><\/h3>\n\n\n\n

We are asked to compute the indefinite integral: \u222bsin\u2061(2t)\u2009dt\\int \\sin(2t) \\, dt\u222bsin(2t)dt<\/p>\n\n\n\n

Solution:<\/strong><\/h4>\n\n\n\n

Using the standard integration rule: \u222bsin\u2061(ax)\u2009dx=\u22121acos\u2061(ax)+C\\int \\sin(ax) \\, dx = -\\frac{1}{a} \\cos(ax) + C\u222bsin(ax)dx=\u2212a1\u200bcos(ax)+C<\/p>\n\n\n\n

Apply this to our integral: \u222bsin\u2061(2t)\u2009dt=\u221212cos\u2061(2t)+C\\int \\sin(2t) \\, dt = -\\frac{1}{2} \\cos(2t) + C\u222bsin(2t)dt=\u221221\u200bcos(2t)+C<\/p>\n\n\n\n

\n\n\n\n

Problem 2: Determine the convolution f(t)=et\u2217cos\u2061(t)f(t) = e^t * \\cos(t)f(t)=et\u2217cos(t)<\/strong><\/h3>\n\n\n\n

We are given two functions: f1(t)=etandf2(t)=cos\u2061(t)f_1(t) = e^t \\quad \\text{and} \\quad f_2(t) = \\cos(t)f1\u200b(t)=etandf2\u200b(t)=cos(t)<\/p>\n\n\n\n

The convolution<\/strong> of two functions is defined as: f(t)=(f1\u2217f2)(t)=\u222b0tf1(\u03c4)f2(t\u2212\u03c4)\u2009d\u03c4f(t) = (f_1 * f_2)(t) = \\int_0^t f_1(\\tau) f_2(t – \\tau) \\, d\\tauf(t)=(f1\u200b\u2217f2\u200b)(t)=\u222b0t\u200bf1\u200b(\u03c4)f2\u200b(t\u2212\u03c4)d\u03c4<\/p>\n\n\n\n

Substitute the functions: f(t)=\u222b0te\u03c4cos\u2061(t\u2212\u03c4)\u2009d\u03c4f(t) = \\int_0^t e^\\tau \\cos(t – \\tau) \\, d\\tauf(t)=\u222b0t\u200be\u03c4cos(t\u2212\u03c4)d\u03c4<\/p>\n\n\n\n

We simplify this using the convolution theorem<\/strong> in the Laplace domain.<\/p>\n\n\n\n

\n\n\n\n

Problem 3: What is the corresponding F(s)F(s)F(s)?<\/strong><\/h3>\n\n\n\n

Using the Laplace transform:<\/p>\n\n\n\n

    \n
  • L{et}=1s\u22121\\mathcal{L}\\{e^t\\} = \\frac{1}{s – 1}L{et}=s\u221211\u200b<\/li>\n\n\n\n
  • L{cos\u2061t}=ss2+1\\mathcal{L}\\{\\cos t\\} = \\frac{s}{s^2 + 1}L{cost}=s2+1s\u200b<\/li>\n<\/ul>\n\n\n\n

    By the Convolution Theorem<\/strong>: L{f(t)}=L{et}\u22c5L{cos\u2061t}\\mathcal{L}\\{f(t)\\} = \\mathcal{L}\\{e^t\\} \\cdot \\mathcal{L}\\{\\cos t\\}L{f(t)}=L{et}\u22c5L{cost} F(s)=1s\u22121\u22c5ss2+1=s(s\u22121)(s2+1)F(s) = \\frac{1}{s – 1} \\cdot \\frac{s}{s^2 + 1} = \\frac{s}{(s – 1)(s^2 + 1)}F(s)=s\u221211\u200b\u22c5s2+1s\u200b=(s\u22121)(s2+1)s\u200b<\/p>\n\n\n\n

    \n\n\n\n

    Textbook-Style Explanation<\/strong><\/h3>\n\n\n\n

    In integral calculus, evaluating trigonometric functions often follows from standard integration formulas. The function sin\u2061(2t)\\sin(2t)sin(2t) is a composite trigonometric function with a linear argument. Applying the substitution rule \u222bsin\u2061(ax)\u2009dx=\u22121acos\u2061(ax)+C\\int \\sin(ax) \\, dx = -\\frac{1}{a} \\cos(ax) + C\u222bsin(ax)dx=\u2212a1\u200bcos(ax)+C, the integral becomes \u221212cos\u2061(2t)+C-\\frac{1}{2} \\cos(2t) + C\u221221\u200bcos(2t)+C. This antiderivative is foundational in analyzing oscillatory systems and arises frequently in physics and engineering.<\/p>\n\n\n\n

    Next, we examine the convolution of two functions, ete^tet and cos\u2061(t)\\cos(t)cos(t). Convolution integrals are essential in systems analysis, particularly in linear time-invariant (LTI) systems where they represent the system\u2019s response to an input. The convolution f(t)=(et\u2217cos\u2061t)(t)f(t) = (e^t * \\cos t)(t)f(t)=(et\u2217cost)(t) is defined by the integral \u222b0te\u03c4cos\u2061(t\u2212\u03c4)\u2009d\u03c4\\int_0^t e^\\tau \\cos(t – \\tau) \\, d\\tau\u222b0t\u200be\u03c4cos(t\u2212\u03c4)d\u03c4, which measures the accumulated effect of e\u03c4e^\\taue\u03c4 scaled by the delayed response cos\u2061(t\u2212\u03c4)\\cos(t – \\tau)cos(t\u2212\u03c4).<\/p>\n\n\n\n

    To evaluate the convolution efficiently, we use the Laplace transform, which converts convolution in the time domain into multiplication in the frequency domain. The Laplace transform of ete^tet is 1s\u22121\\frac{1}{s – 1}s\u221211\u200b, and for cos\u2061(t)\\cos(t)cos(t), it is ss2+1\\frac{s}{s^2 + 1}s2+1s\u200b. Multiplying these gives the transform of the convolution: F(s)=s(s\u22121)(s2+1)F(s) = \\frac{s}{(s – 1)(s^2 + 1)}F(s)=(s\u22121)(s2+1)s\u200b<\/p>\n\n\n\n

    This expression describes the system behavior in the Laplace domain and facilitates further analysis such as inverse transforms, stability analysis, or control design.<\/p>\n\n\n\n

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

    Evaluate the integral “sin 2t d. (). Determine the convolution flt) = e2 COS What is the corresponding F(8)?’ The Correct Answer and Explanation is: Problem 1: Evaluate the integral of sin(2t) We are asked to compute the indefinite integral: \u222bsin\u2061(2t)\u2009dt\\int \\sin(2t) \\, dt\u222bsin(2t)dt Solution: Using the standard integration rule: \u222bsin\u2061(ax)\u2009dx=\u22121acos\u2061(ax)+C\\int \\sin(ax) \\, dx = […]<\/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-233203","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233203","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=233203"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233203\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233203"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233203"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}