   Chapter 8.5, Problem 37E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Integration by Parts In Exercises 33-38, use integration by parts to find the indefinite integral. ∫ t   csc   3 t   cot   3 t   d t

To determine

To calculate: The integral of tcsc3tcot3tdt.

Explanation

Given Information:

The provided integral is tcsc3tcot3tdt.

Formula used:

Write the formula used in integration by parts method.

pqdx=pqdxp(qdx)dx

Here, p=p(x) and q=q(x)

Write the formula of integral of cscucotu.

cscucotudu=cscu+C

Write the formula of integral of cscu.

cscudu=ln|cscu+cotu|+C

Write the formula for derivative of un.

ddxun=nun1

The provided integral is,

tcsc3tcot3tdt

Use the formula of integration by parts to find integral of above expression.

tcsc3tcot3tdt=tcsc3tcot3tdtddtt(csc3tcot3tdt)dt=tcsc3tcot3tdt1(csc3tcot3tdt)dt

Here, p=t and q=csc3tcot3t

Multiply and divide by 3.

tcsc3tcot3tdt=13t(csc3tcot3t)3dtI131((csc3tcot3t)3dt)dtII

Again, multiply and divide by 3 in part II.

tcsc3tcot3tdt=13t(csc3tcot3t)3dt191((csc3tcot3t)3dt)3dt

let u=3t

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