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Chapter 5: Q. 66 (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).
$鈭� \sin h^{2} x \cos h^{2} x . d x$ .

Short Answer

Expert verified

The value of $鈭� \sin h^{2} x \cos h^{2} x . d x$ is, $- \frac{x}{8} + \frac{1}{32} \sin h (4 x) + C$ .

Step by step solution

01

Step 1. Given information

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

02

Step 2. The integration is solved below.

$鈭� \sin h^{2} x \cos h^{2} x . d x = \frac{1}{4} 鈭� \sin h^{2} (2 x) . d x = \frac{1}{4} 鈭� \frac{1}{2} (\cos h (4 x) - 1) . d x = \frac{1}{8} 鈭� (\cos h (4 x) - 1) . d x = \frac{1}{8} 鈭� \cos h (4 x) . d x - \frac{1}{8} 鈭� d x = \frac{1}{8} 鈭� \cos h (4 x) . d x - \frac{x}{8} = \frac{1}{32} 鈭� \cos h (u) . d u - \frac{x}{8} [u = 4 x] = \frac{1}{32} \sin h (u) - \frac{x}{8} + C = - \frac{x}{8} + \frac{1}{32} \sin h (4 x) + C$

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Most popular questions from this chapter

Consider the integral $鈭� \frac{1}{x^{2} \sqrt{1 鈭� x^{2}}} d x$ from the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution $u = \sin^{鈭� 1} 鈦� x$ to solve this integral.

(b) Use the trigonometric substitution $x = \sin 鈦� u$ to solve the integral.

(c) Compare and contrast the two methods used in parts (a) and (b).

Solve the integral: $鈭� \ln \sqrt{x} d x$

Solve given definite integral.

${鈭�}_{4}^{5} 鈥� \frac{1}{x \sqrt{x^{2} + 9}} d x$

Find three integrals in Exercises 21鈥�70 that we can anti-differentiate immediately after algebraic simplification.

Solve the integral: $鈭� x \sin {(x)}^{2} d x$ .

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