Q. 70 Q. Solve each of the integrals i... [FREE SOLUTION]

Chapter 5: Q. 70 (page 429)

Q. Solve each of the integrals in Exercises $27 鈥� 70 .$ Some integrals require integration by parts, and some do not.
$鈭� x^{3} \cos h 2 x d x$ .

Short Answer

Expert verified

The value of $鈭� x^{3} \cos h 2 x d x$ is, $\frac{(4 x^{3} + 6 x) \sin h (2 x) + (- 6 x^{2} - 3) \cos h (2 x)}{8} + C$ .

Step by step solution

Step 1. Given information

$鈭� x^{3} \cos h 2 x d x$ .

Step 2. Now evaluate the given integral.

Integrating by parts, $鈭� f g^{'} = f g - 鈭� f^{'} g$ .

Here, $f = x^{3}, g^{'} = \cos h (2 x) .$

$f^{'} = 3 x^{2}, g = \frac{\sin h (2 x)}{2} 鈭� x^{3} \cos h (2 x) d x = x^{3} \frac{\sin h (2 x)}{2} - 鈭� 3 x^{2} \frac{\sin h (2 x)}{2} d x = x^{3} \frac{\sin h (2 x)}{2} - [\frac{3}{2} 鈭� x^{2} \sin h (2 x) d x] = x^{3} \frac{\sin h (2 x)}{2} - [\frac{3}{2} [\frac{x^{2} \cos h (2 x)}{2} - 鈭� x \cos h (2 x) d x]] = x^{3} \frac{\sin h (2 x)}{2} - \frac{3}{4} x^{2} \cos h (2 x) + \frac{3}{2} 鈭� x \cos h (2 x) d x$

Step 3. Use the Integration by parts method in the required places in the problem.

$鈭� x^{3} \cos h (2 x) d x = x^{3} \frac{\sin h (2 x)}{2} - \frac{3}{4} x^{2} \cos h (2 x) + \frac{3}{2} [x \frac{\sin h (2 x)}{2} - 鈭� \frac{\sin h (2 x)}{2} d x] = x^{3} \frac{\sin h (2 x)}{2} - \frac{3}{4} x^{2} \cos h (2 x) + \frac{3}{2} \frac{x \sin h (2 x)}{2} - \frac{3}{2} 鈭� \frac{\sin h (2 x)}{2} d x = x^{3} \frac{\sin h (2 x)}{2} - \frac{3}{4} x^{2} \cos h (2 x) + \frac{3}{2} \frac{x \sin h (2 x)}{2} - \frac{3}{2} \frac{\cos h (2 x)}{4} = \frac{2 x^{3} \sin h (2 x) + 3 x \sin h (2 x)}{4} + (\frac{- 6 x^{2} \cos h (2 x) - 3 \cos h (2 x)}{8}) = \frac{(4 x^{3} + 6 x) \sin h (2 x)}{8} + (\frac{(- 6 x^{2} - 3) \cos h (2 x)}{8}) = \frac{(4 x^{3} + 6 x) \sin h (2 x) + (- 6 x^{2} - 3) \cos h (2 x)}{8} + C$

Therefore, the required value is, $\frac{(4 x^{3} + 6 x) \sin h (2 x) + (- 6 x^{2} - 3) \cos h (2 x)}{8} + C$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Q. Solve each of the integrals in Exercises $27 鈥� 70 .$ Some integrals require integration by parts, and some do not.
$鈭� x^{3} \cos h 2 x d x$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Now evaluate the given integral.

Step 3. Use the Integration by parts method in the required places in the problem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Calculus

Logic and Functions

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

Q. Solve each of the integrals in Exercises 27鈥�70.Some integrals require integration by parts, and some do not.鈭�x3cosh2xdx.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Now evaluate the given integral.

Step 3. Use the Integration by parts method in the required places in the problem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Calculus

Logic and Functions

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Q. Solve each of the integrals in Exercises $27 鈥� 70 .$ Some integrals require integration by parts, and some do not.
$鈭� x^{3} \cos h 2 x d x$ .