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

The integral of sin(ln(y)) is 1/2 ysin(ln(y))-1/2 ycos(ln(y))+C

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integral of sin(ln(y))

\int sin (ln (y)) d y

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

Solution steps

\int sin (ln (y)) d y

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of sin(ln(y)) ?
The integral of sin(ln(y)) is 1/2 ysin(ln(y))-1/2 ycos(ln(y))+C