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

The integral of (sin(θ)+cos(θ))/(sin(θ)) is θ+ln|sin(θ)|+C

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integral of (sin(θ)+cos(θ))/(sin(θ))

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

θ + ln ∣ sin (θ) ∣ + C

Solution steps

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

Expand

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

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

\int 1 d θ = θ

\int \frac{cos ( θ )}{sin ( θ )} d θ = ln ∣ sin (θ) ∣

= θ + ln ∣ sin (θ) ∣

Add a constant to the solution

= θ + ln ∣ sin (θ) ∣ + C

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What is the integral of (sin(θ)+cos(θ))/(sin(θ)) ?
The integral of (sin(θ)+cos(θ))/(sin(θ)) is θ+ln|sin(θ)|+C