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

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

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integral of (ln(x)sin(ln(x)))/x

\int \frac{ln ( x ) sin ( ln ( x ) )}{x} d x

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

Solution steps

\int \frac{ln ( x ) sin ( ln ( x ) )}{x} d x

Apply u-substitution

= \int u sin (u) d u

Apply Integration By Parts

= - u cos (u) - \int - cos (u) d u

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

= - u cos (u) - (- sin (u))

Substitute back

u = ln (x)

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

Simplify

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

Add a constant to the solution

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

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