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

The integral of (tan(x))/(1+sin^2(x)) is 1/4 ln|2tan^2(x)+1|+C

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

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

\frac{1}{4} ln 2 tan^{2} (x) + 1 + C

Solution steps

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

Apply Integration By Parts

: tan (x) \frac{1}{2} arctan (2 tan (x)) - \int sec^{2} (x) \frac{1}{2} arctan (2 tan (x)) d x

= tan (x) \frac{1}{2} arctan (2 tan (x)) - \int sec^{2} (x) \frac{1}{2} arctan (2 tan (x)) d x

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

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

Simplify

tan (x) \frac{1}{2} arctan (2 tan (x)) - \frac{1}{2} (2 tan (x) arctan (2 tan (x)) - \frac{1}{2} ln 2 tan^{2} (x) + 1) : \frac{1}{4} ln 2 tan^{2} (x) + 1

= \frac{1}{4} ln 2 tan^{2} (x) + 1

Add a constant to the solution

= \frac{1}{4} ln 2 tan^{2} (x) + 1 + C

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What is the integral of (tan(x))/(1+sin^2(x)) ?
The integral of (tan(x))/(1+sin^2(x)) is 1/4 ln|2tan^2(x)+1|+C