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Chapter 5: Q. 55 (page 442)

Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.

鈭�lnxx(1+lnx)2dx

Short Answer

Expert verified

The value is-ln(x)1+ln(x)+ln|1+ln(x)|+C

Step by step solution

01

Step 1. Given Information: 

Given integral :鈭�lnxx(1+lnx)2dx

We want to calculate each of the integrals.

02

Step 2. Calculation:

Applyintegrationbyparts鈭�lnxx(1+lnx)2dx=-lnx1+inx-鈭�-1x(1+ln(x))dx=-ln(x)1+in(x)+鈭�1x(1+ln(x))dx=-ln(x)1+ln(x)+ln|1+ln(x)|+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).

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving鈭�1x2鈭�4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving鈭�1x2鈭�4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving鈭�1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving鈭�x2+4鈭�5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2鈭�a2.

(f) True or False: Trigonometric substitution doesn鈥檛 solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

鈭�xx2+1dx

Solve the integral:鈭�xexdx

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+6x鈭�2
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