Evaluate the integral: int_((1)/(8))^((1)/(4))sin^(2)(2x)dx

Rewrite using power-reduction formula: Using the power-reduction formula, we can rewrite the integral as follows: int_((1)/(8))^((1)/(4))sin^(2)(2x)dx = int_((1)/(8))^((1)/(4))(1/2 - 1/2*cos(4x))dx This simplifies the integral into two separate integrals: (1/2) * int_((1)/(8))^((1)/(4))dx - (1/2) * int_((1)/(8))^((1)/(4))cos(4x)dxIntegrate separate terms: Now we will integrate each term separately. The integral of 1 with respect to x is simply x, and the integral of cos(4x) with respect to x is (1/4)sin(4x). So, the integral becomes: (1/2) * [x]_((1)/(8))^((1)/(4)) - (1/2) * [(1/4)sin(4x)]_((1)/(8))^((1)/(4))Evaluate first term: We will now evaluate the definite integrals at the upper and lower limits of integration. For the first term: (1/2) * [x]_((1)/(8))^((1)/(4)) = (1/2) * [(1/4) - (1/8)] = (1/2) * (1/8) = 1/16 For the second term: (1/2) * [(1/4)sin(4x)]_((1)/(8))^((1)/(4)) = (1/8) * [sin(1) - sin(1/2)]Calculate sin values: Now we will calculate the values of sin(1) and sin(1/2) to find the exact value of the second term. sin(1) and sin(1/2) are values that can be looked up or calculated using a calculator. For the sake of this problem, we will assume we have the exact values: sin(1) ≈ 0.84147 sin(1/2) ≈ 0.47943 So, the second term becomes: (1/8) * [0.84147 - 0.47943] = (1/8) * 0.36204 ≈ 0.04526Find final value: Adding the results of the two terms together gives us the final value of the integral: 1/16 + 0.04526 ≈ 0.06276 + 0.04526 ≈ 0.10802 This is the exact, simplified answer to the integral.

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Evaluate the integral: int_((1)/(8))^((1)/(4))sin^(2)(2x)dx

Evaluate the integral: $\int_{\frac{1}{8}}^{\frac{1}{4}} \sin ^{2}(2 x) d x$

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate the integral:

\int_{\frac{1}{8}}^{\frac{1}{4}} \sin ^{2}(2 x) d x

Rewrite using power-reduction formula: Using the power-reduction formula, we can rewrite the integral as follows: $\newline$ $\int_{(1)/(8)}^{(1)/(4)}\sin^{2}(2x)\,dx = \int_{(1)/(8)}^{(1)/(4)}(\frac{1}{2} - \frac{1}{2}\cos(4x))\,dx$ $\newline$ This simplifies the integral into two separate integrals: $\newline$ $(\frac{1}{2}) \int_{(1)/(8)}^{(1)/(4)}\,dx - (\frac{1}{2}) \int_{(1)/(8)}^{(1)/(4)}\cos(4x)\,dx$
Integrate separate terms: Now we will integrate each term separately. The integral of $1$ with respect to $x$ is simply $x$ , and the integral of $\cos(4x)$ with respect to $x$ is $(1/4)\sin(4x)$ . So, the integral becomes: $($1$/$2$) \cdot [x]_{($1$)/($8$)}^{($1$)/($4$)} - ($1$/$2$) \cdot [($1$/$4$)\sin($4$x)]_{($1$)/($8$)}^{($1$)/($4$)}
Evaluate first term: We will now evaluate the definite integrals at the upper and lower limits of integration.$\newline$For the first term:$\newline$$(1/2) \times [x]_{(1)/(8)}^{(1)/(4)} = (1/2) \times [(1/4) - (1/8)] = (1/2) \times (1/8) = 1/16$$\newline$For the second term:$\newline$$(1/2) \times [(1/4)\sin(4x)]_{(1)/(8)}^{(1)/(4)} = (1/8) \times [\sin(1) - \sin(1/2)]$
Calculate sin values: Now we will calculate the values of $\sin(1)$ and $\sin(\frac{1}{2})$ to find the exact value of the second term.$\newline$$\sin(1)$ and $\sin(\frac{1}{2})$ are values that can be looked up or calculated using a calculator. For the sake of this problem, we will assume we have the exact values:$\newline$$\sin(1) \approx 0.84147$$\newline$$\sin(\frac{1}{2}) \approx 0.47943$$\newline$So, the second term becomes:$\newline$$(\frac{1}{8}) \times [0.84147 - 0.47943] = (\frac{1}{8}) \times 0.36204 \approx 0.04526$
Find final value: Adding the results of the two terms together gives us the final value of the integral:$\newline$$\frac{1}{16} + 0.04526 \approx 0.06276 + 0.04526 \approx 0.10802$$\newline$This is the exact, simplified answer to the integral.