   Chapter 8.3, Problem 10E

Chapter
Section
Textbook Problem

Finding an Indefinite Integral Involving Sine and Cosine In Exercises 3-14, find the indefinite integral. ∫ cos 5 t sin t d t

To determine

To calculate: The value of indefinite integral given as, cos5tsintdt.

Explanation

Given:

The function is cos5tsintdt.

Formula used:

Integration of xn is given as,

xndx=xn+1n+1+C

Derivative of sinx with respect to x is given as,

ddx(sinx)=cosx

Calculation:

Let the given integral be equal to I,

I=cos5tsintdt

Simplify it to a form in which the power rule can be used,

I=cos5tsintdt=(cos2t)2(sint)12(cost)dt=(1sin2t)2(sint)12(cost)dt

For the integral which involves the powers of sine and cosine,

Put, x=sint and differentiate both side with respect to t as,

dx=costdt

Substitute the value, sint=x,

I=(1sin2t)2(sint)12(cost)dt=(1x2)2(x)12dx=(12x2+x4)(x)

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