What is the integral of (1+tan(θ))^2 ?

The integral of (1+tan(θ))^2 is -2ln|cos(θ)|+tan(θ)+C

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integral of (1+tan(θ))^2

\int (1 + tan (θ))^{2} d θ

- 2 ln ∣ cos (θ) ∣ + tan (θ) + C

Solution steps

\int (1 + tan (θ))^{2} d θ

Rewrite using trig identities

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

Expand

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

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

\int 1 d θ = θ

\int \frac{2 sin ( θ )}{cos ( θ )} d θ = - 2 ln ∣ cos (θ) ∣

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

= θ - 2 ln ∣ cos (θ) ∣ + tan (θ) - θ

Simplify

= - 2 ln ∣ cos (θ) ∣ + tan (θ)

Add a constant to the solution

= - 2 ln ∣ cos (θ) ∣ + tan (θ) + C

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What is the integral of (1+tan(θ))^2 ?
The integral of (1+tan(θ))^2 is -2ln|cos(θ)|+tan(θ)+C