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

The integral of sin(ln(w)) is 1/2 wsin(ln(w))-1/2 wcos(ln(w))+C

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integral of sin(ln(w))

\int sin (ln (w)) d w

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

Solution steps

\int sin (ln (w)) d w

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of sin(ln(w)) ?
The integral of sin(ln(w)) is 1/2 wsin(ln(w))-1/2 wcos(ln(w))+C