What is the integral of ln((cos(θ))^{tan(θ)}) ?

The integral of ln((cos(θ))^{tan(θ)}) is -(ln^2(cos(θ)))/2+C

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integral of ln((cos(θ))^{tan(θ)})

\int ln ((cos (θ))^{t a n (θ)}) d θ

- \frac{ln ^{2} ( cos ( θ ) )}{2} + C

Solution steps

\int ln ((cos (θ))^{t a n (θ)}) d θ

Apply u-substitution

: \int - \frac{ln ( u )}{u} d u

= \int - \frac{ln ( u )}{u} d u

Take the constant out:

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

= - \int \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= - \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= - \frac{v ^{2}}{2}

Substitute back

= - \frac{ln ^{2} ( cos ( θ ) )}{2}

Add a constant to the solution

= - \frac{ln ^{2} ( cos ( θ ) )}{2} + C

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What is the integral of ln((cos(θ))^{tan(θ)}) ?
The integral of ln((cos(θ))^{tan(θ)}) is -(ln^2(cos(θ)))/2+C