What is the integral of (tan(x))(ln(cos(x))) ?

The integral of (tan(x))(ln(cos(x))) is -(ln^2(cos(x)))/2+C

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integral of (tan(x))(ln(cos(x)))

\int (tan (x)) (ln (cos (x))) d x

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

Solution steps

\int tan (x) ln (cos (x)) d x

Apply u-substitution

= \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

Apply the Power Rule

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

Substitute back

= - \frac{ln ^{2} ( cos ( x ) )}{2}

Add a constant to the solution

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

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