Q. 74 Solve each of the definite integ... [FREE SOLUTION]

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Chapter 5: Q. 74 (page 452)

Solve each of the definite integrals in Exercises 67鈥�76.
${鈭�}_{蟺 / 4}^{蟺 / 2} c s c x c o t^{3} x d x$ .

Short Answer

Expert verified

The answer is $\frac{2 - \sqrt{2}}{3}$ .

Step by step solution

Step 1. Given Information.

The integral is ${鈭�}_{蟺 / 4}^{蟺 / 2} c s c x c o t^{3} x d x$ .

Step 2. Explanation.

Simplify the integral by substituting $c s c x = \frac{1}{\sin x}$ and $c o t x = \frac{\cos x}{\sin x}$ .

localid="1649227085459" ${鈭�}_{蟺 / 4}^{蟺 / 2} c s c x c o t^{3} x d x = {鈭�}_{蟺 / 4}^{蟺 / 2} \frac{\cos^{3} x}{\sin^{4} x} d x$

Use Pythagorean identity $\cos^{2} x = 1 - \sin^{2} x$ .

${鈭�}_{蟺 / 4}^{蟺 / 2} \frac{\cos^{3} x}{\sin^{4} x} d x = {鈭�}_{蟺 / 4}^{蟺 / 2} \frac{1 - \sin^{2} x}{\sin^{4} x} \cos x d x$

Substitute $u = \sin x$ and differentiate it.

$d u = \cos x d x$ .

So, the integral becomes as follows,

role="math" localid="1649227537451" ${鈭�}_{\frac{1}{\sqrt{2}}}^{1} \frac{1 - u^{2}}{u^{4}} d u$

Step 3. Calculation.

Further simplify the integral.

${鈭�}_{\frac{1}{\sqrt{2}}}^{1} \frac{1 - u^{2}}{u^{4}} d u = {鈭�}_{\frac{1}{\sqrt{2}}}^{1} \frac{1}{u^{4}} d u - {鈭�}_{\frac{1}{\sqrt{2}}}^{1} \frac{1}{u^{2}} d u = {[- \frac{1}{3 u^{3}}]}_{\frac{1}{\sqrt{2}}}^{1} - {[- \frac{1}{u}]}_{\frac{1}{\sqrt{2}}}^{1} = \frac{- 1 + 2 \sqrt{2}}{3} + 1 - \sqrt{2} = \frac{2 - \sqrt{2}}{3}$

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91影视

Solve each of the definite integrals in Exercises 67鈥�76.
${鈭�}_{蟺 / 4}^{蟺 / 2} c s c x c o t^{3} x d x$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Calculation.

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91影视

Solve each of the definite integrals in Exercises 67鈥�76.鈭�蟺/4蟺/2cscxcot3xdx.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Calculus

Geometry

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Solve each of the definite integrals in Exercises 67鈥�76.
${鈭�}_{蟺 / 4}^{蟺 / 2} c s c x c o t^{3} x d x$ .