What is the integral of sin(ln(u)) ?

The integral of sin(ln(u)) is 1/2 usin(ln(u))-1/2 ucos(ln(u))+C

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integral of sin(ln(u))

\int sin (ln (u)) d u

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

Solution steps

\int sin (ln (u)) d u

Apply Integration By Parts

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

\int cos (ln (u)) d u = \frac{1}{2} u sin (ln (u)) + \frac{1}{2} u cos (ln (u))

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of sin(ln(u)) ?
The integral of sin(ln(u)) is 1/2 usin(ln(u))-1/2 ucos(ln(u))+C