What is the integral of ((cos(θ)+tan(θ)))/(cos^2(θ)) ?

The integral of ((cos(θ)+tan(θ)))/(cos^2(θ)) is ln|tan(θ)+sec(θ)|+(tan^2(θ))/2+C

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integral of ((cos(θ)+tan(θ)))/(cos^2(θ))

\int \frac{( cos ( θ ) + tan ( θ ) )}{cos ^{2} ( θ )} d θ

ln ∣ tan (θ) + sec (θ) ∣ + \frac{tan ^{2} ( θ )}{2} + C

Solution steps

\int \frac{cos ( θ ) + tan ( θ )}{cos ^{2} ( θ )} d θ

Expand

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

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

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

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

= ln ∣ tan (θ) + sec (θ) ∣ + \frac{tan ^{2} ( θ )}{2}

Add a constant to the solution

= ln ∣ tan (θ) + sec (θ) ∣ + \frac{tan ^{2} ( θ )}{2} + C

What is the integral of ((cos(θ)+tan(θ)))/(cos^2(θ)) ?
The integral of ((cos(θ)+tan(θ)))/(cos^2(θ)) is ln|tan(θ)+sec(θ)|+(tan^2(θ))/2+C