What is the integral of (sin^2(x))/((1+sin^2(x))) ?

The integral of (sin^2(x))/((1+sin^2(x))) is -1/(sqrt(2))arctan(sqrt(2)tan(x))+x+C

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integral of (sin^2(x))/((1+sin^2(x)))

integral

\frac{sin ^{2} ( x )}{( 1 + sin ^{2} ( x ) )}

- \frac{1}{2} arctan (2 tan (x)) + x + C

Solution steps

\int \frac{sin ^{2} ( x )}{1 + sin ^{2} ( x )} d x

\frac{sin ^{2} ( x )}{1 + sin ^{2} ( x )} = - \frac{1}{1 + sin ^{2} ( x )} + 1

= \int - \frac{1}{1 + sin ^{2} ( x )} + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{1 + sin ^{2} ( x )} d x + \int 1 d x

\int \frac{1}{1 + sin ^{2} ( x )} d x = \frac{1}{2} arctan (2 tan (x))

\int 1 d x = x

= - \frac{1}{2} arctan (2 tan (x)) + x

Add a constant to the solution

= - \frac{1}{2} arctan (2 tan (x)) + x + C

\frac{x ( x + 1 )}{2}

(\frac{1}{x}) (cos (x) - sin (x))

x cos (x) cos (2 x)

x^{- 6}

\frac{( x + 7 )}{( x + 9 )}

What is the integral of (sin^2(x))/((1+sin^2(x))) ?
The integral of (sin^2(x))/((1+sin^2(x))) is -1/(sqrt(2))arctan(sqrt(2)tan(x))+x+C