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Prove that \[\large\int_{0}^{\frac{\pi}{4}} \left(\dfrac{x}{x\sin x+\cos x}\right)^2 dx=\dfrac{4-\pi}{4+\pi}.\]

Note by Adarsh Kumar
10 months, 1 week ago

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By Differentiation - Quotient Rule , the integrand might be in the form of \( \dfrac {v u' - u v'}{v^2} \), so we can make the assumption that \(v = x\sin(x) + \cos(x) \).

Now we just want to find the unknown coefficients for \(u = A x\cos(x) + Bx \sin(x) + C \sin(x)+ D \cos(x) + Ex \) such that \(vu' - uv' = x^2 \). Solving this gives \(u = \sin(x) - x\cos(x) \).

Checking back, the indefinite integral of \( \dfrac{x^2}{x\sin(x) + \cos(x)} \) is indeed \( \dfrac{\sin(x) - x\cos(x)}{x\sin(x) + \cos(x)} \). Substituting the limits give the desired answer. Pi Han Goh · 10 months, 1 week ago

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@Pi Han Goh Challenge student note: nice solution. U have used the trick perfectly. Aditya Kumar · 10 months, 1 week ago

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@Pi Han Goh Yes,thank you!Actually this problem is from a book,the book too had a similar solution,I just wanted a different one,an easier one! Adarsh Kumar · 10 months, 1 week ago

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@Adarsh Kumar Please don't use square brackets if it is not a gif. It confuses. Aditya Kumar · 10 months, 1 week ago

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@Aditya Kumar Ooops!Sorry! Adarsh Kumar · 10 months, 1 week ago

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I will post this one later. Got it. Mardokay Mosazghi · 10 months, 1 week ago

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