Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

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

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

$d v = \cos x d x v = \int \cos x d x v = \sin x$

The formula of integration by parts is $\int u d v = u v - \int v d u$ .

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

Step 3. Integration by parts.

We have,

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

and

$d v = \sin x d x v = \int \sin x d x v = - \cos x$

So,

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

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Calculus

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Short Answer

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

Step by Step Solution

Step 1. Given information.

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

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