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Math Help - Integrate:

  1. #1
    Member great_math's Avatar
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    Integrate:

    \int \frac{1}{\sin^2 x+\sin x+1}\ dx
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  2. #2
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    Quote Originally Posted by great_math View Post
    \int \frac{1}{\sin^2 x+\sin x+1}\ dx
    Consider the integral,
    \int \frac{dx}{\sin^2 (2x) + \sin (2x ) + 1} = \int \frac{dx}{\left( \frac{2\tan x}{1+\tan^2x} \right)^2 + \left( \frac{2 \tan x}{1+\tan^2 x} \right) + 1}

    Which simplifies to,
    \int \frac{(1+\tan^2 x)(1+\tan^2 x) dx}{4\tan^2 x + 2\tan x + 1} dx = \int \frac{(1+\tan^2 x) (\sec^2 x)}{4\tan^2 x + 2\tan x + 1}

    Now let t=\tan x \implies t' = \sec^2 x by substitution we get,
    \int \frac{1+t^2}{4t^2 + 2t + 1}dt

    Can you continue?
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