Q. 53 Solve each of the integrals in E... [FREE SOLUTION]

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Chapter 5: Q. 53 (page 429)

Solve each of the integrals in Exercises 27鈥�70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$鈭� x^{3} \cos x d x$

Short Answer

Expert verified

The value is $x^{3} \sin x + 3 x^{2} \cos x - 6 x \sin x - 6 \cos x + C$ .

Step by step solution

Step 1. Given information.

The given integral is $鈭� x^{3} \cos x d x$ .

Step 2. Substitution.

Now,

$u = x^{3}, d v = \cos x d x, d u = 3 x^{2} d x, v = 鈭� \cos x d x = \sin x 鈭� x^{3} \cos x d x = (x^{3} \sin x - 鈭� \sin (x) 路 3 x^{2} d x) = x^{3} \sin x - 3 鈭� x^{2} \sin (x) d x let u = x^{2}, d v = \sin x d x . x^{3} \sin x - 3 鈭� x^{2} \sin (x) d x = = x^{3} \sin x - 3 [x^{2} 路 (- \cos x) - 鈭� 2 x (- \cos (x)) d x] = x^{3} \sin x - 3 [- x^{2} \cos x + 2 鈭� x \cos x d x] = x^{3} \sin x + 3 x^{2} \cos x - 6 鈭� x \cos x d x$

Step 3. Value of the integral.

Again,

$u = x, d v = \cos x d x v = 鈭� \cos x d x = \sin x x^{3} \sin x + 3 x^{2} \cos x - 6 鈭� x \cos x d x = = x^{3} \sin x + 3 x^{2} \cos x - 6 [x \sin x - 鈭� \sin x d x] = x^{3} \sin x + 3 x^{2} \cos x - 6 x \sin x + 6 (- \cos x) + C = x^{3} \sin x + 3 x^{2} \cos x - 6 x \sin x - 6 \cos x + C$

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91影视

Solve each of the integrals in Exercises 27鈥�70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$鈭� x^{3} \cos x d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitution.

Step 3. Value of the integral.

One App. One Place for Learning.

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91影视

Solve each of the integrals in Exercises 27鈥�70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)鈭�x3cosxdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitution.

Step 3. Value of the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Pure Maths

Geometry

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 27鈥�70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$鈭� x^{3} \cos x d x$