What is the integral of xsin^2(ln(x)) ?

The integral of xsin^2(ln(x)) is 1/2 x^2sin^2(ln(x))-1/8 (x^2sin(2ln(x))-x^2cos(2ln(x)))+C

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integral of xsin^2(ln(x))

\int x sin^{2} (ln (x)) d x

\frac{1}{2} x^{2} sin^{2} (ln (x)) - \frac{1}{8} (x^{2} sin (2 ln (x)) - x^{2} cos (2 ln (x))) + C

Solution steps

\int x sin^{2} (ln (x)) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} sin^{2} (ln (x)) - \int \frac{1}{2} x sin (2 ln (x)) d x

= \frac{1}{2} x^{2} sin^{2} (ln (x)) - \int \frac{1}{2} x sin (2 ln (x)) d x

\int \frac{1}{2} x sin (2 ln (x)) d x = \frac{1}{8} (x^{2} sin (2 ln (x)) - x^{2} cos (2 ln (x)))

= \frac{1}{2} x^{2} sin^{2} (ln (x)) - \frac{1}{8} (x^{2} sin (2 ln (x)) - x^{2} cos (2 ln (x)))

Add a constant to the solution

= \frac{1}{2} x^{2} sin^{2} (ln (x)) - \frac{1}{8} (x^{2} sin (2 ln (x)) - x^{2} cos (2 ln (x))) + C

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What is the integral of xsin^2(ln(x)) ?
The integral of xsin^2(ln(x)) is 1/2 x^2sin^2(ln(x))-1/8 (x^2sin(2ln(x))-x^2cos(2ln(x)))+C