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Chapter 5: Q 29. (page 429)

Solve the integral: $鈭� x \ln (x) d x$

Short Answer

Expert verified

The required answer is $\frac{x^{2}}{2} \ln x - \frac{x^{2}}{4} + c$ .

Step by step solution

01

Step 1. Given information.

We have given integral is $鈭� x \ln (x) d x$ .

02

Step 2. Solve the integration by parts .

We have, $u = \ln x, d v = x d x$ .

$u = \ln x d u = \frac{1}{x} d x$

and

$d v = x d x v = 鈭� x d x v = \frac{x^{2}}{2}$

The formula of integration by parts is $鈭� u d v = u v - 鈭� v d u$

$鈭� x \ln (x) d x = \ln (x) \frac{x^{2}}{2} - 鈭� \frac{x^{2}}{2} 脳 \frac{1}{x} d x = \frac{x^{2}}{2} \ln (x) - 鈭� \frac{x}{2} d x = \frac{x^{2}}{2} \ln (x) - \frac{x^{2}}{4} + c$

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Most popular questions from this chapter

Solve the integral: $鈭� x \ln \sqrt{x} d x$

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

$鈭� \sin u d u$

Find three integrals in Exercises 27鈥�70 for which either algebra or u-substitution is a better strategy than integration by parts.

Find three integrals in Exercises 39鈥�74 that can be solved without using trigonometric substitution.

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.

$鈭� \frac{\sqrt{9 - x^{2}}}{x} d x$

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