Q. 46 Solve each of the integrals in E... [FREE SOLUTION]

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Chapter 5: Q. 46 (page 452)

Solve each of the integrals in Exercises 21鈥�66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions.)
$鈭� \csc^{4} x \cot^{2} x d x$

Short Answer

Expert verified

The solution of the integral is $- \frac{\cot^{3} x}{3} - \frac{\cot^{5} x}{5} + C$

Step by step solution

Step 1. Given Information

The given integral is $鈭� \csc^{4} x \cot^{2} x d x$ .

Step 2. Rewrite and substitute

Use the trigonometric identities to rewrite the integral as follows:

$鈭� \csc^{4} x \cot^{2} x d x = 鈭� \csc^{2} x (1 + \cot^{2} x) \cot^{2} x d x$

Substitute $\cot x = u$ into the integral.
So, $- \csc^{2} x d x = d u$ . Substitute and simplify the integral.

$鈭� \csc^{2} x (1 + \cot^{2} x) \cot^{2} x d x = - 鈭� (1 + u^{2}) u^{2} d x = - 鈭� u^{2} + u^{4} d x$

Step 3. Integrate

Perform integration on the obtained integral.

$- 鈭� u^{2} + u^{4} d x = - \frac{u^{3}}{3} - \frac{u^{5}}{5} + C$

Substitute $u = \cot (x)$ into the obtained integral to find the solution.
So, the value of the integral is localid="1649008393273" role="math" $- \frac{\cot^{3} x}{3} - \frac{\cot^{5} x}{5} + C$

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91影视

Solve each of the integrals in Exercises 21鈥�66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions.)
$鈭� \csc^{4} x \cot^{2} x d x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Rewrite and substitute

Step 3. Integrate

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91影视

Solve each of the integrals in Exercises 21鈥�66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions.)鈭�csc4xcot2xdx

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Rewrite and substitute

Step 3. Integrate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Calculus

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 21鈥�66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions.)
$鈭� \csc^{4} x \cot^{2} x d x$