What is the integral of sin(log_{e}(x)) ?

The integral of sin(log_{e}(x)) is 1/2 xsin(ln(x))-1/2 xcos(ln(x))+C

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integral of sin(log_{e}(x))

\int sin (lo g_{e} (x)) d x

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

Solution steps

\int sin (lo g_{e} (x)) d x

Simplify

lo g_{e} (x) : ln (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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\int (7 x^{3} + 5 x^{2} + \frac{9}{x ^{2}} + \frac{2}{a ^{2}}) d x

What is the integral of sin(log_{e}(x)) ?
The integral of sin(log_{e}(x)) is 1/2 xsin(ln(x))-1/2 xcos(ln(x))+C