What is the integral of [sin(ln(x))] ?

The integral of [sin(ln(x))] is 1/2 xsin(ln(x))-1/2 xcos(ln(x))+C

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integral of [sin(ln(x))]

\int [sin (ln (x))] d x

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

Solution steps

\int sin (ln (x)) d x

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of [sin(ln(x))] ?
The integral of [sin(ln(x))] is 1/2 xsin(ln(x))-1/2 xcos(ln(x))+C