What is the integral of 2+tan^2(θ) ?

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

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integral of 2+tan^2(θ)

\int 2 + tan^{2} (θ) d θ

tan (θ) + θ + C

Solution steps

\int 2 + tan^{2} (θ) d θ

Rewrite using trig identities

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

\int sec^{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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