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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int 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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

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