What is the integral of (cos(θ))/(1+cos(θ)) ?

The integral of (cos(θ))/(1+cos(θ)) is -csc(θ)+cot(θ)+θ+C

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integral of (cos(θ))/(1+cos(θ))

\int \frac{cos ( θ )}{1 + cos ( θ )} d θ

- csc (θ) + cot (θ) + θ + C

Solution steps

\int \frac{cos ( θ )}{1 + cos ( θ )} d θ

Multiply by the conjugate of

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

= \int \frac{cos ( θ )}{\frac{s i n ^{2} ( θ )}{1 - c o s ( θ )}} d θ

Expand

\frac{cos ( θ )}{\frac{s i n ^{2} ( θ )}{1 - c o s ( θ )}} : \frac{cos ( θ )}{sin ^{2} ( θ )} - \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

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

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{cos ( θ )}{sin ^{2} ( θ )} d θ - \int \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} d θ

\int \frac{cos ( θ )}{sin ^{2} ( θ )} d θ = - csc (θ)

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

= - csc (θ) - (- cot (θ) - θ)

Simplify

= - csc (θ) + cot (θ) + θ

Add a constant to the solution

= - csc (θ) + cot (θ) + θ + C

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What is the integral of (cos(θ))/(1+cos(θ)) ?
The integral of (cos(θ))/(1+cos(θ)) is -csc(θ)+cot(θ)+θ+C