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

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

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

integral

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

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

Solution steps

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

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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

tan^{4} (x) sec^{3} (x)

2 - (x - 2)^{2}

e^{- 2} t

\frac{( 2 x ^{2} - 1 )}{( x - x ^{3} )}

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

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