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Chapter 5: Q. 70 (page 452)

Solve each of the definite integrals in Exercises 67鈥�76.
${鈭�}_{- 蟺}^{蟺} \sin^{2} (x) d x$

Short Answer

Expert verified

The solution of the integral is $蟺$ .

Step by step solution

01

Step 1. Given Information

The given integral is ${鈭�}_{- 蟺}^{蟺} \sin^{2} (x) d x$ .

02

Step 2. Rewrite

Rewrite the integral using trigonometric identities as follows:

$\begin{array}{rcl} {鈭�}_{- 蟺}^{蟺} \sin^{2} (x) d x & = & {鈭�}_{- 蟺}^{蟺} \frac{1 - \cos (2 x)}{2} d x \\ = & \frac{1}{2} 路 {鈭�}_{- 蟺}^{蟺} 1 - \cos (2 x) d x \end{array}$

03

Step 3. Integrate

Integrate the obtained integral by using the integration formulas.

$\begin{array}{rcl} \frac{1}{2} 路 {鈭�}_{- 蟺}^{蟺} 1 - \cos (2 x) d x & = & \frac{1}{2} ({鈭�}_{- 蟺}^{蟺} 1 d x - {鈭�}_{- 蟺}^{蟺} \cos (2 x) d x) \\ = & \frac{1}{2} {(x - \frac{\sin (2 x)}{2})}_{- 蟺}^{蟺} \\ = & \frac{1}{2} (2 蟺 - 0) \\ = & 蟺 \end{array}$

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Most popular questions from this chapter

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