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

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

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

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

- cot (x) + ln (sin^{2} (x)) + C

Solution steps

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

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

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

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

= - cot (x) + ln (sin^{2} (x))

Add a constant to the solution

= - cot (x) + ln (sin^{2} (x)) + C

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