Integrate: `int (cos x) /[4+sin^2x] dx`

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lemjay | High School Teacher | (Level 3) Senior Educator

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`int (cosx)/(4+sin^2x) dx = int 1/(4+sin^2x)cosx dx`

Use u-substitution.


   `u = sin x`

Differentiate u.

 `du = cos x dx`

Then, replace x with u variable.

`int 1/(4+sin^2x)cosx dx = int 1/(4+u^2)du`

To integrate, use the integral of rational functions which is` int 1/(a^2 + u^2) du = 1/a tan^(-1) u/a + C` .

`int 1/(2^2+u^2)du = 1/2 tan^(-1)u/2 + C`

Then, return back the variable x. Substitute `u` with `sin x` .

`= 1/2 tan^(-1) (sinx)/2 + C`

Answer: `int (cosx)/(4+sin^2x) dx = 1/2 tan^(-1) (sinx)/2 + C`