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Chapter 5: Q. 66 (page 418)

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�cotxln(sinx)dx

Short Answer

Expert verified

The solution of the given integral is 鈭�cotxln(sinx)dx=12(ln(sinx))2+C.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

鈭�cotxln(sinx)dx

02

Step 2. Using the substitution method.

u=ln(sinx)dudx=1sinx路cosxdudx=cosxsinxdudx=cotxdu=cotxdx

03

Step 3. This substitution changes the integral into 

鈭�cotxln(sinx)dx=鈭�udu鈭�cotxln(sinx)dx=u1+11+1+C鈭�cotxln(sinx)dx=u22+C鈭�cotxln(sinx)dx=12u2+C鈭�cotxln(sinx)dx=12(ln(sinx))2+C

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Most popular questions from this chapter

Problem Zero: Read the section and make your own summary of the material.

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.

2x2鈭�4x+1

Why don鈥檛 we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution x=atan鈦�uwhen we see x2+a2, even though we can鈥檛 use the substitution x=asin鈦�uunless the integrand involves the square root ofa2鈭�x2? (Hint: Think about domains.)

Find three integrals in Exercises 21鈥�70 that we can anti-differentiate immediately after algebraic simplification.

Solve the integral:鈭�xlnxdx
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