Q. 64 Solve each of the integrals in E... [FREE SOLUTION]

Chapter 5: Q. 64 (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.)
$鈭� \sin h^{5} (x) \cos h^{2} (x) d x$

Short Answer

Expert verified

The solution is $\begin{array}{rcl} \frac{\cos h^{7} (x)}{7} + \frac{\cos h^{3}}{(x) 3} - \frac{2}{5} \cos h^{5} (x) + C \end{array}$

Step by step solution

Step 1. Given Information

The given integral is $鈭� \sin h^{5} (x) \cos h^{2} (x) d x$ .

Step 2. Rewrite and substitute

Use the trigonometric identities to rewrite the integral as follows:

$\begin{array}{rcl} 鈭� \sin h^{5} (x) \cos h^{2} (x) d x & = & 鈭� \sin h (x) \sin h^{4} (x) \cos h^{2} (x) d x \\ = & 鈭� \sin h (x) {(\cos h^{2} (x) - 1)}^{2} \cos h^{2} (x) d x \end{array}$

Assume that $\cos h (x) = u$ . So, $\sin h (x) d x = d u$ .
Substitute the value into the integral an simplify.

$\begin{array}{rcl} 鈭� \sin h (x) {(\cos h^{2} (x) - 1)}^{2} \cos h^{2} (x) d x & = & 鈭� {(u^{2} - 1)}^{2} u^{2} d u = 鈭� (u^{4} + 1 - 2 u^{2}) u^{2} d u \\ = & 鈭� (u^{6} + u^{2} - 2 u^{4}) d u \end{array}$

Step 3. Integrate

Integrate the obtained integral and then substitute $u = \cos h (x)$ to find the solution.

localid="1649132385014" $\begin{array}{rcl} 鈭� (u^{6} + u^{2} - 2 u^{4}) d u & = & \frac{u^{7}}{7} + \frac{u^{3}}{3} - 2 \frac{u^{5}}{5} + C \\ = & \frac{\cos h^{7} (x)}{7} + \frac{\cos h^{3}}{(x) 3} - \frac{2}{5} \cos h^{5} (x) + C \end{array}$

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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.)
$鈭� \sin h^{5} (x) \cos h^{2} (x) d x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Rewrite and substitute

Step 3. Integrate

One App. One Place for Learning.

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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�sinh5(x)cosh2(x)dx

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

Mechanics Maths

Geometry

Decision Maths

Pure Maths

Applied Mathematics

Calculus

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.)
$鈭� \sin h^{5} (x) \cos h^{2} (x) d x$