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

The integral of (1+cos^2(θ))/(cos^2(θ)) is tan(θ)+θ+C

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integral of (1+cos^2(θ))/(cos^2(θ))

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

tan (θ) + θ + C

Solution steps

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

Expand

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

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

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

\int 1 d θ = θ

= tan (θ) + θ

Add a constant to the solution

= tan (θ) + θ + C

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