Evaluate the integral: int_((-pi)/(8))^((pi)/(8))sin^(2)(2x)dx

Apply power-reduction formula: Use the power-reduction formula for sine, which states that sin^2(x) = (1 - cos(2x))/2. This will simplify the integral. sin^2(2x) = (1 - cos(4x))/2 The integral becomes: int_((-pi)/(8))^((pi)/(8))sin^(2)(2x)dx = int_((-pi)/(8))^((pi)/(8))(1/2 - 1/2 cos(4x))dxSplit into two integrals: Split the integral into two separate integrals. int_((-pi)/(8))^((pi)/(8))(1/2 - 1/2 cos(4x))dx = 1/2 int_((-pi)/(8))^((pi)/(8))dx - 1/2 int_((-pi)/(8))^((pi)/(8))cos(4x)dxEvaluate constant integral: Evaluate the first integral, which is a constant with respect to x. 1/2 int_((-pi)/(8))^((pi)/(8))dx = 1/2 [x]_((-pi)/(8))^((pi)/(8)) = 1/2 [(pi/8) - (-pi/8)] = 1/2 * (pi/4) = pi/8Integrate cosine function: Evaluate the second integral, which involves the cosine function. 1/2 int_((-pi)/(8))^((pi)/(8))cos(4x)dx To integrate cos(4x), we use the substitution u = 4x, which gives us du = 4dx or dx = du/4.Change limits with substitution: Change the limits of integration according to the substitution u = 4x. When x = -pi/8, u = -pi/2. When x = pi/8, u = pi/2. The integral becomes: 1/2 * 1/4 int_((-pi)/(2))^((pi)/(2))cos(u)duEvaluate integral of cosine: Evaluate the integral of cos(u) from -pi/2 to pi/2. 1/2 * 1/4 int_((-pi)/(2))^((pi)/(2))cos(u)du = 1/8 [sin(u)]_((-pi)/(2))^((pi)/(2)) = 1/8 [sin(pi/2) - sin(-pi/2)] = 1/8 * (1 - (-1)) = 1/8 * 2 = 1/4Combine results for final answer: Combine the results from Step 3 and Step 6 to get the final answer. The integral of sin^2(2x) from (-pi/8) to (pi/8) is: pi/8 - 1/4Simplify final answer: Simplify the final answer. pi/8 - 1/4 = (pi - 2)/8

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Evaluate the integral: int_((-pi)/(8))^((pi)/(8))sin^(2)(2x)dx

Evaluate the integral: $\int_{\frac{-\pi}{8}}^{\frac{\pi}{8}} \sin ^{2}(2 x) d x$

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Evaluate definite integrals using the chain rule

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Q. Evaluate the integral:

\int_{\frac{-\pi}{8}}^{\frac{\pi}{8}} \sin ^{2}(2 x) d x

Apply power-reduction formula: Use the power-reduction formula for sine, which states that $\sin^2(x) = \frac{1 - \cos(2x)}{2}$ . This will simplify the integral. $\newline$ $\sin^2(2x) = \frac{1 - \cos(4x)}{2}$ $\newline$ The integral becomes: $\newline$ $\int_{(-\pi)/(8)}^{(\pi)/(8)}\sin^{2}(2x)dx = \int_{(-\pi)/(8)}^{(\pi)/(8)}\left(\frac{1}{2} - \frac{1}{2} \cos(4x)\right)dx$
Split into two integrals: Split the integral into two separate integrals. \int_{(-\pi)/($8$)}^{(\pi)/($8$)}(\frac{$1$}{$2$} - \frac{$1$}{$2$} \cos(\(4x))dx = \frac{ $1$ }{ $2$ } \int_{(-\pi)/( $8$ )}^{(\pi)/( $8$ )}dx - \frac{ $1$ }{ $2$ } \int_{(-\pi)/( $8$ )}^{(\pi)/( $8$ )}\cos( $4$ x)dx
Evaluate constant integral: Evaluate the first integral, which is a constant with respect to $x$ . $\frac{1}{2} \int_{(-\pi)/(8)}^{(\pi)/(8)}dx = \frac{1}{2} [x]_{(-\pi)/(8)}^{(\pi)/(8)} = \frac{1}{2} [(\pi/8) - (-\pi/8)] = \frac{1}{2} \cdot (\pi/4) = \pi/8$
Integrate cosine function: Evaluate the second integral, which involves the cosine function. $\newline$ $\frac{1}{2}$ $\int_{(-\pi)/(8)}^{(\pi)/(8)}\cos(4x)\,dx$ $\newline$ To integrate $\cos(4x)$ , we use the substitution $u = 4x$ , which gives us $du = 4dx$ or $dx = \frac{du}{4}$ .
Change limits with substitution: Change the limits of integration according to the substitution $u = 4x$ . $\newline$ When $x = -\frac{\pi}{8}$ , $u = -\frac{\pi}{2}$ . $\newline$ When $x = \frac{\pi}{8}$ , $u = \frac{\pi}{2}$ . $\newline$ The integral becomes: $\newline$ $\frac{1}{2} \times \frac{1}{4} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos(u)\,du$
Evaluate integral of cosine: Evaluate the integral of $\cos(u)$ from $-\pi/2$ to $\pi/2$ . $\frac{1}{2} \times \frac{1}{4} \int_{(-\pi)/(2)}^{(\pi)/(2)}\cos(u)\,du = \frac{1}{8} [\sin(u)]_{(-\pi)/(2)}^{(\pi)/(2)} = \frac{1}{8} [\sin(\pi/2) - \sin(-\pi/2)] = \frac{1}{8} \times (1 - (-1)) = \frac{1}{8} \times 2 = \frac{1}{4}$
Combine results for final answer: Combine the results from Step $3$ and Step $6$ to get the final answer. $\newline$ The integral of $\sin^2(2x)$ from $(-\pi/8)$ to $(\pi/8)$ is: $\newline$ $\pi/8 - 1/4$
Simplify final answer: Simplify the final answer. $\frac{\pi}{8} - \frac{1}{4} = \frac{\pi - 2}{8}$