What is the integral of sin(2ln(x)) ?

The integral of sin(2ln(x)) is 1/5 xsin(2ln(x))-2/5 xcos(2ln(x))+C

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integral of sin(2ln(x))

\int sin (2 ln (x)) d x

\frac{1}{5} x sin (2 ln (x)) - \frac{2}{5} x cos (2 ln (x)) + C

Solution steps

\int sin (2 ln (x)) d x

Apply Integration By Parts

: x sin (2 ln (x)) - \int 2 cos (2 ln (x)) d x

= x sin (2 ln (x)) - \int 2 cos (2 ln (x)) d x

\int 2 cos (2 ln (x)) d x = 2 (\frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x)))

= x sin (2 ln (x)) - 2 (\frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x)))

Simplify

x sin (2 ln (x)) - 2 (\frac{2}{5} x sin (2 ln (x)) + \frac{1}{5} x cos (2 ln (x))) : \frac{1}{5} x sin (2 ln (x)) - \frac{2}{5} x cos (2 ln (x))

= \frac{1}{5} x sin (2 ln (x)) - \frac{2}{5} x cos (2 ln (x))

Add a constant to the solution

= \frac{1}{5} x sin (2 ln (x)) - \frac{2}{5} x cos (2 ln (x)) + C

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What is the integral of sin(2ln(x)) ?
The integral of sin(2ln(x)) is 1/5 xsin(2ln(x))-2/5 xcos(2ln(x))+C