What is the integral of (sin^3(θ)+1)/(sin^2(θ)) ?

The integral of (sin^3(θ)+1)/(sin^2(θ)) is -cos(θ)-cot(θ)+C

Popular >

integral of (sin^3(θ)+1)/(sin^2(θ))

\int \frac{sin ^{3} ( θ ) + 1}{sin ^{2} ( θ )} d θ

- cos (θ) - cot (θ) + C

Solution steps

\int \frac{sin ^{3} ( θ ) + 1}{sin ^{2} ( θ )} d θ

Expand

\frac{sin ^{3} ( θ ) + 1}{sin ^{2} ( θ )} : sin (θ) + \frac{1}{sin ^{2} ( θ )}

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

\int sin (θ) d θ = - cos (θ)

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

= - cos (θ) - cot (θ)

Add a constant to the solution

= - cos (θ) - cot (θ) + C

\int \frac{1}{3 x + 9} d x

\int (x^{5} - 5^{x} + 5 x - \frac{5}{x}) d x

\int 2 y^{- 1} d y

\int (\frac{6 x + 1}{3 3 x + 2}) d x

\int \frac{1}{5 y + 8} d y

What is the integral of (sin^3(θ)+1)/(sin^2(θ)) ?
The integral of (sin^3(θ)+1)/(sin^2(θ)) is -cos(θ)-cot(θ)+C