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Reduction for integrals of tan^n x or cot^n x using I(n) + I(n + 2)

This type of question, asked 3 times: 2021 Paper 4, 13 Apr 2023, 22 Jan 2025. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 4

    64
    If In=∫π/4π/2cot⁡nxdxI_{n}=\int_{\pi/4}^{\pi/2} \cot^{n}xdx, then:

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  2. Question 2

    13 Apr 2023

    170
    The value of e−π4+∫0π4e−xtan⁡50xdx∫0π4e−x(tan⁡49x+tan⁡51x)dx\frac{e^{-\frac{\pi}{4}}+\int_{0}^{\frac{\pi}{4}} e^{- x}\tan^{50}xdx}{\int_{0}^{\frac{\pi}{4}} e^{- x}\left( \tan^{49}x +\tan^{51}x \right)dx} is

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  3. Question 3

    22 Jan 2025

    65
    Let for f(x)=7tan⁡8x+7tan⁡6x−3tan⁡4x−3tan⁡2xf(x) = 7\tan^{8}x + 7\tan^{6}x - 3\tan^{4}x - 3\tan^{2}x, I1=∫0π/4f(x)dxI_{1}=\int_{0}^{\pi/4} f(x)dx and I2=∫0π/4xf(x)dxI_{2}=\int_{0}^{\pi/4} xf(x)dx. Then 7I1+12I27I_{1}+ 12I_{2} is equal to :

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