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

The integral of (ln(cos(x)))sin(x) is cos(x)-cos(x)ln(cos(x))+C

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integral of (ln(cos(x)))sin(x)

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

cos (x) - cos (x) ln (cos (x)) + C

Solution steps

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

Apply Integration By Parts

: - \int sin (x) d x - cos (x) ln (cos (x))

= - \int sin (x) d x - cos (x) ln (cos (x))

\int sin (x) d x = - cos (x)

= - (- cos (x)) - cos (x) ln (cos (x))

Simplify

= cos (x) - cos (x) ln (cos (x))

Add a constant to the solution

= cos (x) - cos (x) ln (cos (x)) + C

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What is the integral of (ln(cos(x)))sin(x) ?
The integral of (ln(cos(x)))sin(x) is cos(x)-cos(x)ln(cos(x))+C