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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

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