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Chapter 5: Q 32. (page 429)

Solve the integral: $鈭� x^{2} \cos x d x$ .

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

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

02

Step 2. Solve the integration by parts .

We have, $u = x^{2} d v = \cos x d x$

$u = x^{2} d u = 2 x d x$

and

$d v = \cos x d x v = 鈭� \cos x d x v = \sin x$

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

$鈭� x^{2} \cos x d x = x^{2} \sin x - 鈭� 2 x \sin x d x = x^{2} \sin x - 2 鈭� x \sin x d x$

03

Step 3. Integration by parts.

We have,

$u = x, d v = \sin x d x u = x d u = d x$

and

$d v = \sin x d x v = 鈭� \sin x d x v = - \cos x$

So,

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

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Most popular questions from this chapter

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

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

Solve $鈭� \frac{x^{2}}{1 + x^{2}} d x$ the following two ways:

(a) with the substitution $u = \tan^{- 1} x;$

(b) with the trigonometric substitution x = tan u.

Solve the integral: $鈭� \frac{x^{2}}{e^{x}} d x$

Solve the integral: $鈭� \frac{x^{2} + 1}{e^{x}} d x$

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