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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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