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Chapter 5: Q 40. (page 429)

Solve the integral: $鈭� x \ln (x^{2}) d x$

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

We have given integral is $鈭� x \ln (x^{2}) d x$ .

02

Step 2. Solve the integration by parts .

We have,

$u = \ln (x) d u = \frac{d x}{x}$

and

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

The formula of integration by parts is $鈭� u d v = u v - 鈭� v d u$
$鈭� x \ln (x^{2}) d x = 2 (\ln (x) \frac{x^{2}}{2} - 鈭� \frac{x^{2}}{2} 脳 \frac{1}{x} d x) = 2 (\ln (x) \frac{x^{2}}{2} - 鈭� \frac{x}{2} d x) = x^{2} \ln (x) - \frac{x^{2}}{2} + c$

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