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Chapter 5: Q. 40 (page 417)

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

鈭�cos(lnx)xdx

Short Answer

Expert verified

The solution of the given integral is 鈭�cos(lnx)xdx=sin(Inx)+C.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

鈭�cos(lnx)xdx

02

Step 2. Solving the given integral using substitution method. 

Let

u=lnxdudx=1xdu=1xdx

03

Step 3. This substitution changes the integral into 

鈭�cos(lnx)xdx=鈭�cosudu鈭�cos(lnx)xdx=sinu+C鈭�cos(lnx)xdx=sin(Inx)+C

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Most popular questions from this chapter

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.

x2-5x+1

Explain why using trigonometric substitution with x=atanuoften involves a triangle with side lengths a and x and hypotenuse of lengthx2+a2

Solve the integral:鈭�x2cosxdx.

Explain why, if x=atanu, then x2+a2=asecu. Your explanation should include a discussion of domains and absolute values.

Find three integrals in Exercises 39鈥�74 that can be solved by using a trigonometric substitution of the form x=atanu.
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