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Chapter 5: Q. 58 (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�1xlnxdx

Short Answer

Expert verified

The solution of the given integral is 鈭�1xlnxdx=ln(lnx)+C.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

鈭�1xlnxdx

02

Step 2. Using the substitution method.

Let

u=lnxdudx=1xdu=1xdx
03

Step 3. This substitution changes the integral into 

鈭�1xlnxdx=鈭�1udu鈭�1xlnxdx=lnu+C鈭�1xlnxdx=ln(lnx)+C

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Most popular questions from this chapter

Solve the integral:鈭�xcosxdx.

Solve the integral

鈭�4-xx-2dx

Explain why 鈭�2xx2+1dxand 鈭�1xlnxdxare essentially the same integral after a change of variables.

Why doesn鈥檛 the definite integral鈭�231-x2dx make sense? (Hint: Think about domains.)

Consider the integral 鈭�sinxcosxdx.

(a) Solve this integral by using u-substitution with u=sinx and du=cosxdx.

(b) Solve the integral another way, using u-substitution with u=cosx and du=鈭�sinxdx.

(c) How must your two answers be related? Use algebra to prove this relationship.
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