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Chapter 5: Q 31. (page 429)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

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

02

Step 2. Solve the integration by parts .

We have, $u = x d u = d x$

and

$d v = \sin^{2} x d x v = 鈭� \sin^{2} x d x v = \frac{x}{2} - \frac{\sin 2 x}{4}$

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

$鈭� x \sin {(x)}^{2} d x = x [\frac{x}{2} - \frac{\sin 2 x}{4}] - 鈭� \frac{x}{2} - \frac{\sin 2 x}{4} d x = \frac{x^{2}}{2} - \frac{x \sin 2 x}{4} - \frac{x^{2}}{4} + 鈭� \frac{\sin 2 x}{4} d x = - \frac{\cos x^{2}}{2} + c$

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