Chapter 5: Q. 70 (page 429)

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

Short Answer

Expert verified

The value of $\int x^{3} \cos h 2 x d x$ 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

$\int x^{3} \cos h 2 x d x$ .

Step 2. Now evaluate the given integral.

Integrating by parts, $\int f g^{'} = f g - \int f^{'} g$ .

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

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

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

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

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

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

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.

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Q. Solve each of the integrals in Exercises 27–70.Some integrals require integration by parts, and some do not.∫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

Pure Maths

Logic and Functions

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Study anywhere. Anytime. Across all devices.

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