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Chapter 5: Q 21. (page 452)

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

Short Answer

Expert verified

Answer is $\begin{array}{rcl} \frac{x}{2} - \frac{\sin 6 x}{12} + C \end{array}$

Step by step solution

Step 1. Given information

Integral is $鈭� \sin^{2} 3 x d x$

Step 2. Explanation

Consider the integral $鈭� \sin^{2} 3 x d x$

Use the formula: $\sin^{2} y = \frac{1 - \cos 2 y}{2}$

So,

$\begin{array}{rcl} 鈭� \sin^{2} 3 x d x & = & 鈭� \frac{1 - \cos 6 x}{2} d x \\ = & \frac{x}{2} - \frac{\sin 6 x}{12} + C \end{array}$

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Most popular questions from this chapter

Explain why it makes sense to try the trigonometric substitution $x = \sec 鈦� u$ if an integrand involves the expression $\sqrt{x^{2} 鈭� 1}$

Solve given definite integrals.

${鈭�}_{0}^{4} x \sqrt{x^{2} + 4} d x$

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

$鈭� \frac{1}{\sqrt{u}} d u$

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x^{2} + 6 x 鈭� 2$

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x^{2} 鈭� 4 x 鈭� 8$

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Solve the integral.鈭�sin23xdx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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