What is the integral of 2θsec^2(θ) ?

The integral of 2θsec^2(θ) is 2(θtan(θ)+ln|cos(θ)|)+C

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integral of 2θsec^2(θ)

\int 2 θ sec^{2} (θ) d θ

2 (θ tan (θ) + ln ∣ cos (θ) ∣) + C

Solution steps

\int 2 θ sec^{2} (θ) d θ

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int θ sec^{2} (θ) d θ

Apply Integration By Parts

: θ tan (θ) - \int tan (θ) d θ

= 2 (θ tan (θ) - \int tan (θ) d θ)

\int tan (θ) d θ = - ln ∣ cos (θ) ∣

= 2 (θ tan (θ) - (- ln ∣ cos (θ) ∣))

Simplify

= 2 (θ tan (θ) + ln ∣ cos (θ) ∣)

Add a constant to the solution

= 2 (θ tan (θ) + ln ∣ cos (θ) ∣) + C

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What is the integral of 2θsec^2(θ) ?
The integral of 2θsec^2(θ) is 2(θtan(θ)+ln|cos(θ)|)+C