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Chapter 5: Q 41. (page 452)

Solve the following integral.
$鈭� \sin^{2} 3 x \cos^{2} 3 x d x$

Short Answer

Expert verified

Answer is $\frac{x}{8} - \frac{\sin 12 x}{96} + C$

Step by step solution

01

Step 1. Given information

An integral is $鈭� \sin^{2} 3 x \cos^{2} 3 x d x$

02

Step 2. Explanation

Let $u = 3 x$

$\begin{array}{rcl} 鈭� \sin^{2} 3 x \cos^{2} 3 x d x & = & \frac{1}{3} 鈭� \sin^{2} u \cos^{2} u d u \\ = & \frac{1}{12} 鈭� \sin^{2} 2 u d u \\ = & \frac{1}{24} 鈭� 1 - \cos 4 u d u \\ = & \frac{1}{24} (u - \frac{\sin 4 u}{4}) + C \\ = & \frac{1}{24} (3 x - \frac{\sin 12 x}{4}) + C \\ = & \frac{x}{8} - \frac{\sin 12 x}{96} + C \end{array}$

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Most popular questions from this chapter

Solve $鈭� \frac{x}{4 + x^{2}} d x$ the following two ways:

(a) with the substitution $u = 4 + x^{2};$

(b) with the trigonometric substitution x = 2 tan u.

For each integral in Exercises 5鈥�8, write down three integrals that will have that form after a substitution of variables.

$鈭� e^{u} d u$

Solve the integral

$鈭� \frac{1}{\sqrt{x^{2} + 9}} d x$

For each integral in Exercises 5鈥�8, write down three integrals that will have that form after a substitution of variables.

$鈭� \frac{1}{\sqrt{u}} d u$

Find three integrals in Exercises 27鈥�70 for which either algebra or u-substitution is a better strategy than integration by parts.

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