Evaluate the trigonomnetric integral∫(sint)5(cost)3dt

Name: Evaluate the trigonomnetric integral∫(sint)5(cost)3dt
Uploaded: 2024-03-20T06:46:55.000Z
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Description: Evaluate the trigonomnetric integral∫(sint)5(cost)3dt

Given, ∫sint5cost3dt We have to evaluate the integral

Answered: Evaluate the trigonomnetric integral…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.2: Trigonometric Functions Of Angles

Problem 7E

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Question

Evaluate the trigonomnetric integral

∫(sint)⁵(cost)³dt

Expert Solution

Step 1

Given,

$\int {(\sin t)}^{5} {(\cos t)}^{3} d t$

We have to evaluate the integral

Step 2

We have $\int {(\sin t)}^{5} {(\cos t)}^{3} d t$

Let's assume

$x = \cos t$

then, $\frac{d x}{d t} = - \sin t$ or $- d x = \sin t d t$

Now substituting the values in $\int {(\sin t)}^{5} {(\cos t)}^{3} d t$

$\int {(\sin t)}^{5} {(\cos t)}^{3} d t = \int {(\sin t)}^{4} {(\cos t)}^{3} \sin t d t$

$\int {(\sin t)}^{5} {(\cos t)}^{3} d t = - \int {(\sin^{2} t)}^{2} {(x)}^{3} d x$

$\int {(\sin t)}^{5} {(\cos t)}^{3} d t = - \int {(1 - \cos^{2} t)}^{2} {(x)}^{3} d x$ $(b e c a u s e \sin^{2} t + \cos^{2} t = 1)$

$\int {(\sin t)}^{5} {(\cos t)}^{3} d t = - \int {(1 - x^{2})}^{2} {(x)}^{3} d x$

$\int {(\sin t)}^{5} {(\cos t)}^{3} d t = - \int (1 - 2 x^{2} + x^{4}) x^{3} d x$

$\int {(\sin t)}^{5} {(\cos t)}^{3} d t = - \int (x^{3} - 2 x^{2 + 3} + x^{4 + 3}) d x$

$\int {(\sin t)}^{5} {(\cos t)}^{3} d t = - \int (x^{3} - 2 x^{5} + x^{7}) d x$

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