Q. 49 Solve each of the integrals in E... [FREE SOLUTION]

Chapter 5: Q. 49 (page 452)

Solve each of the integrals in Exercises $21 - 66$ . Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions).
role="math" localid="1649522213988" $鈭� c s c x c o t^{2} x d x$ .

Short Answer

Expert verified

The value of the integral $鈭� c s c x c o t^{2} x d x$ is, $- \frac{1}{2} c s c x . c o t x + \frac{1}{2} \ln |c s c x + c o t x| + C$ .

Step by step solution

Step 1. Given information

$鈭� c s c x c o t^{2} x d x$ .

Step 2. The integration is solved below.

In $鈭� c s c x c o t^{2} x d x$ , first let us try to find the value for $c s c x . c o t^{2} x$ .

$c s c x . c o t^{2} x = c s c x (c s c^{2} x - 1) = c s c^{3} x - c s c x$

Now find the integral value for $c s c^{3} x$ .

Let $u = c s c x, d u = - c o t x c s c x . d x$

$v = - c o t x, d v = c s c^{2} x$ .

Now apply the integrating by parts rule.

鈭� c s c^{3} x . d x = - c s c x . c o t x - 鈭� c o t^{2} x . c s c x . d x = - c s c x . c o t x - 鈭� c s c x . (c s c^{2} x - 1) d x = - c s c x . c o t x - 鈭� (c s c^{3} x - c s c x) d x = - c s c x . c o t x - 鈭� c s c^{3} x . d x + 鈭� c s c x . d x 2 鈭� c s c^{3} x . d x = - c s c x . c o t x + 鈭� c s c x . d x

Step 3. Now apply the theorem 5.17 in the obtained equation.

$2 鈭� c s c^{3} x . d x = - c s c x . c o t x - \ln |c s c x + c o t x| + C 鈭� c s c^{3} x . d x = - \frac{1}{2} c s c x . c o t x - \frac{1}{2} \ln |c s c x + c o t x| + C$

Now substitute the known values and also apply theorem $5.17$ in $鈭� (c s c^{3} x - c s c x) d x$ .

role="math" localid="1649527490064" $鈭� (c s c^{3} x - c s c x) d x = - \frac{1}{2} c s c x . c o t x - \frac{1}{2} \ln |c s c x + c o t x| - (- \ln |c s c x + c o t x|) + C = - \frac{1}{2} c s c x . c o t x + \frac{1}{2} \ln |c s c x + c o t x| + C$

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91影视

Solve each of the integrals in Exercises $21 - 66$ . Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions).
role="math" localid="1649522213988" $鈭� c s c x c o t^{2} x d x$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. The integration is solved below.

Step 3. Now apply the theorem 5.17 in the obtained equation.

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91影视

Solve each of the integrals in Exercises 21-66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions).role="math" localid="1649522213988" 鈭�cscxcot2xdx.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The integration is solved below.

Step 3. Now apply the theorem 5.17 in the obtained equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Probability and Statistics

Applied Mathematics

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises $21 - 66$ . Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions).
role="math" localid="1649522213988" $鈭� c s c x c o t^{2} x d x$ .