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

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

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integral of (1+cos(2t))/(sin^2(t))

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

- 2 cot (t) - 2 t + C

Solution steps

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

Expand

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

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

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} ( t )} d t + \int \frac{cos ( 2 t )}{sin ^{2} ( t )} d t

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

\int \frac{cos ( 2 t )}{sin ^{2} ( t )} d t = - cot (t) - 2 t

= - cot (t) - cot (t) - 2 t

Simplify

= - 2 cot (t) - 2 t

Add a constant to the solution

= - 2 cot (t) - 2 t + C

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