integral of (tan(θ)+sec(θ))^2

解答

\int (tan (θ) + sec (θ))^{2} d θ

- θ + 2 tan (θ) + 2 sec (θ) + C

求解步骤

\int (tan (θ) + sec (θ))^{2} d θ

乘开

(tan (θ) + sec (θ))^{2} : tan^{2} (θ) + 2 tan (θ) sec (θ) + sec^{2} (θ)

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

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int tan^{2} (θ) d θ = - θ + tan (θ)

\int 2 tan (θ) sec (θ) d θ = 2 sec (θ)

\int sec^{2} (θ) d θ = tan (θ)

= - θ + tan (θ) + 2 sec (θ) + tan (θ)

化简

= - θ + 2 tan (θ) + 2 sec (θ)

解答补常数

= - θ + 2 tan (θ) + 2 sec (θ) + C

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