What is the integral of 2cos(2x)ln(cos(x)) ?

The integral of 2cos(2x)ln(cos(x)) is ln(cos(x))sin(2x)+x-1/2 sin(2x)+C

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integral of 2cos(2x)ln(cos(x))

\int 2 cos (2 x) ln (cos (x)) d x

ln (cos (x)) sin (2 x) + x - \frac{1}{2} sin (2 x) + C

Solution steps

\int 2 cos (2 x) ln (cos (x)) d x

Take the constant out:

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

= 2 \cdot \int cos (2 x) ln (cos (x)) d x

Apply Integration By Parts

: \frac{1}{2} ln (cos (x)) sin (2 x) - \int - \frac{1}{2} tan (x) sin (2 x) d x

= 2 (\frac{1}{2} ln (cos (x)) sin (2 x) - \int - \frac{1}{2} tan (x) sin (2 x) d x)

\int - \frac{1}{2} tan (x) sin (2 x) d x = - \frac{1}{2} (x - \frac{1}{2} sin (2 x))

= 2 (\frac{1}{2} ln (cos (x)) sin (2 x) - (- \frac{1}{2} (x - \frac{1}{2} sin (2 x))))

Simplify

2 (\frac{1}{2} ln (cos (x)) sin (2 x) - (- \frac{1}{2} (x - \frac{1}{2} sin (2 x)))) : ln (cos (x)) sin (2 x) + x - \frac{1}{2} sin (2 x)

= ln (cos (x)) sin (2 x) + x - \frac{1}{2} sin (2 x)

Add a constant to the solution

= ln (cos (x)) sin (2 x) + x - \frac{1}{2} sin (2 x) + C

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What is the integral of 2cos(2x)ln(cos(x)) ?
The integral of 2cos(2x)ln(cos(x)) is ln(cos(x))sin(2x)+x-1/2 sin(2x)+C