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

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

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integral of (sin(θ)+cos(θ))/(cos(θ))

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

- ln ∣ cos (θ) ∣ + θ + C

Solution steps

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

Expand

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

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

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

\int 1 d θ = θ

= - ln ∣ cos (θ) ∣ + θ

Add a constant to the solution

= - ln ∣ cos (θ) ∣ + θ + C

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