$I_{2}=\int_{0}^{\frac{\pi}{2}} \sin ^{4} x d x$

Q: $I_{2}=\int_{0}^{\frac{\pi}{2}} \sin ^{4} x d x$

∻Solution by Steps∻ ‖ step 1 ⋮ To find the integral of $ \sin^4(x) $ from $ 0 $ to $ \frac{\pi}{2} $, we use the power-reduction formula for even powers of sine. ‖ ‖ step 2 ⋮ The integral of $ \sin^4(x) $ can be expressed as $ \int_0^{\frac{\pi}{2}} \frac{3}{8} - \frac{1}{4} \cos(2x) + \frac{1}{32} \cos(4x) \, dx $. ‖ ‖ step 3 ⋮ Integrating term by term, we get $ \left[\frac{3x}{8} - \frac{1}{8} \sin(2x) + \frac{1}{128} \sin(4x)\right]_0^{\frac{\pi}{2}} $. ‖ ‖ step 4 ⋮ Evaluating the antiderivative at the bounds $ 0 $ and $ \frac{\pi}{2} $, we find $ \left(\frac{3\pi}{16} - 0 + 0\right) - \left(0 - 0 + 0\right) $. ‖ ‖ step 5 ⋮ The result of the definite integral is $ \frac{3\pi}{16} $. ‖ ∻Answer∻ ⚹ $ \frac{3\pi}{16} $ ⚹ ∻Key Concept∻ ⚹Power-Reduction Formula for Even Powers of Sine⚹ ∻Explanation∻ ⚹The integral of $ \sin^4(x) $ from $ 0 $ to $ \frac{\pi}{2} $ can be computed by using the power-reduction formula to rewrite the integral in terms of cosine, which simplifies the integration process.⚹◊What is the value of $I_{2}$?⍭ Generate me a similar question◊

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the integral of $ \sin^4(x) $ from $ 0 $ to $ \frac{\pi}{2} $, we use the power-reduction formula for even powers of sine. ‖ ‖ step 2 ⋮ The integral of $ \sin^4(x) $ can be expressed as $ \int_0^{\frac{\pi}{2}} \frac{3}{8} - \frac{1}{4} \cos(2x) + \frac{1}{32} \cos(4x) \, dx $. ‖ ‖ step 3 ⋮ Integrating term by term, we get $ \left[\frac{3x}{8} - \frac{1}{8} \sin(2x) + \frac{1}{128} \sin(4x)\right]_0^{\frac{\pi}{2}} $. ‖ ‖ step 4 ⋮ Evaluating the antiderivative at the bounds $ 0 $ and $ \frac{\pi}{2} $, we find $ \left(\frac{3\pi}{16} - 0 + 0\right) - \left(0 - 0 + 0\right) $. ‖ ‖ step 5 ⋮ The result of the definite integral is $ \frac{3\pi}{16} $. ‖ ∻Answer∻ ⚹ $ \frac{3\pi}{16} $ ⚹ ∻Key Concept∻ ⚹Power-Reduction Formula for Even Powers of Sine⚹ ∻Explanation∻ ⚹The integral of $ \sin^4(x) $ from $ 0 $ to $ \frac{\pi}{2} $ can be computed by using the power-reduction formula to rewrite the integral in terms of cosine, which simplifies the integration process.⚹◊What is the value of $I_{2}$?⍭ Generate me a similar question◊

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