What is the integral of ln(x)cos(3x) ?

The integral of ln(x)cos(3x) is 1/3 ln(x)sin(3x)-1/3 \Si(3x)+C

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integral of ln(x)cos(3x)

\int ln (x) cos (3 x) d x

\frac{1}{3} ln (x) sin (3 x) - \frac{1}{3} Si (3 x) + C

Solution steps

\int ln (x) cos (3 x) d x

Apply Integration By Parts

: \frac{1}{3} ln (x) sin (3 x) - \int \frac{sin ( 3 x )}{3 x} d x

= \frac{1}{3} ln (x) sin (3 x) - \int \frac{sin ( 3 x )}{3 x} d x

\int \frac{sin ( 3 x )}{3 x} d x = \frac{1}{3} Si (3 x)

= \frac{1}{3} ln (x) sin (3 x) - \frac{1}{3} Si (3 x)

Add a constant to the solution

= \frac{1}{3} ln (x) sin (3 x) - \frac{1}{3} Si (3 x) + C

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What is the integral of ln(x)cos(3x) ?
The integral of ln(x)cos(3x) is 1/3 ln(x)sin(3x)-1/3 \Si(3x)+C