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Math Help - Four trigonometric integrals.

  1. #1
    Junior Member
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    Four trigonometric integrals.

    Hello! I'm having some troubles solving this integrals so your help would be much appreciated.

    1. \int \frac{dx}{tan^8x}

    2. \int \frac{dx}{\sqrt[4]{sin^3xcos^5x}}

    3. \int \frac{dx}{(sinx+cosx)^2}

    4. \frac{\sqrt{tgx}dx}{sinxcosx}
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  2. #2
    MHF Contributor red_dog's Avatar
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    1) I_n=\int\frac{dx}{\tan^nx}=\int\cot^nxdx=\int\cot^  {n-2}x\cdot\cot^2xdx=\int\cot^{n-2}x\left(\frac{1}{\sin^2x}-1\right)dx=

    =-\int\cot^{n-2}x\cdot(\cot x)'dx-I_{n-2}=-\frac{\cot^{n-1}x}{n-1}-I_{n-2}

    3) \sin x+\cos x=\sin x+\sin\left(\frac{\pi}{2}-x\right)=2\sin\frac{\pi}{4}\cos\left(x-\frac{\pi}{4}\right)=\sqrt{2}\cos\left(x-\frac{\pi}{4}\right)

    Then, \int\frac{dx}{2\cos^2\left(x-\frac{\pi}{4}\right)}=\frac{1}{2}\tan\left(x-\frac{\pi}{4}\right)+C
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  3. #3
    Senior Member Peritus's Avatar
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    4. use the substitution


    u = \sqrt {\tan x}
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