What is the integral of sin(θ)tan(θ) ?

The integral of sin(θ)tan(θ) is ln|tan(θ)+sec(θ)|-sin(θ)+C

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integral of sin(θ)tan(θ)

\int sin (θ) tan (θ) d θ

ln ∣ tan (θ) + sec (θ) ∣ - sin (θ) + C

Solution steps

\int sin (θ) tan (θ) d θ

Rewrite using trig identities

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

Expand

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

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

\int \frac{1}{cos ( θ )} d θ = ln ∣ tan (θ) + sec (θ) ∣

\int cos (θ) d θ = sin (θ)

= ln ∣ tan (θ) + sec (θ) ∣ - sin (θ)

Add a constant to the solution

= ln ∣ tan (θ) + sec (θ) ∣ - sin (θ) + C

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