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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

= - (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

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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