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Chapter 5: Q 62. (page 429)

Solve the integral:鈭�sinx3xdx

Short Answer

Expert verified

The required answer is-ln3sinx-cosx3x1+ln23+c

Step by step solution

01

Step 1. Given information. 

We have given integral is鈭�sinx3xdx.

02

Step 2. Solve the integration by parts . 

We have,

u=sinxdu=cosxdx

and

dv=13xdxv=鈭�13xdxv=-13xln(3)

The formula of integration by parts is 鈭�udv=uv-鈭�vdu.

role="math" localid="1648894863931" 鈭�Sinx3xdx=Sinx-13xln(3)-鈭�cosx-13xln3dx=-鈭�-cosx3xln3dx-sin(x)3xln(3)

03

Step 3. Integration by parts. 

We have,

u=cosxdu=-sinxdx

and

dv=13xdxv=鈭�13xdxv=-13xln(3)

So,

localid="1648895035220" -鈭�-cosx3xln3dx-sin(x)3xln(3)=-ln(3)sin(x)-cosx3x(1+ln2(3))+c

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Most popular questions from this chapter

Find three integrals in Exercises 21鈥�70 in which the denominator of the integrand is a good choice for a substitution u(x).

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution withx=asinuor x=atanu? Why do we need to think about the unit circle after trigonometric substitution with x=asecu?

For each integral in Exercises 5鈥�8, write down three integrals that will have that form after a substitution of variables.

鈭�eudu

Find three integrals in Exercises 27鈥�70 for which a good strategy is to use integration by parts with u=xand dv the remaining part.

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x鈭�3x2
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