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

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

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

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

x - cot (x) + C

Solution steps

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

Simplify

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

= \int 1 + csc^{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 csc^{2} (x) d x

\int 1 d x = x

\int csc^{2} (x) d x = - cot (x)

= x - cot (x)

Add a constant to the solution

= x - cot (x) + C

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\int \frac{1 + x}{x ^{2} - 4 x + 3} d x

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