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

The integral of x^2cos(ln(x)) is 3/10 x^3cos(ln(x))+1/10 x^3sin(ln(x))+C

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integral of x^2cos(ln(x))

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

\frac{3}{10} x^{3} cos (ln (x)) + \frac{1}{10} x^{3} sin (ln (x)) + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{3} x^{3} cos (ln (x)) - \int - \frac{1}{3} x^{2} sin (ln (x)) d x

= \frac{1}{3} x^{3} cos (ln (x)) - \int - \frac{1}{3} x^{2} sin (ln (x)) d x

\int - \frac{1}{3} x^{2} sin (ln (x)) d x = - \frac{1}{30} (- x^{3} cos (ln (x)) + 3 x^{3} sin (ln (x)))

= \frac{1}{3} x^{3} cos (ln (x)) - (- \frac{1}{30} (- x^{3} cos (ln (x)) + 3 x^{3} sin (ln (x))))

Simplify

\frac{1}{3} x^{3} cos (ln (x)) - (- \frac{1}{30} (- x^{3} cos (ln (x)) + 3 x^{3} sin (ln (x)))) : \frac{3}{10} x^{3} cos (ln (x)) + \frac{1}{10} x^{3} sin (ln (x))

= \frac{3}{10} x^{3} cos (ln (x)) + \frac{1}{10} x^{3} sin (ln (x))

Add a constant to the solution

= \frac{3}{10} x^{3} cos (ln (x)) + \frac{1}{10} x^{3} sin (ln (x)) + C

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What is the integral of x^2cos(ln(x)) ?
The integral of x^2cos(ln(x)) is 3/10 x^3cos(ln(x))+1/10 x^3sin(ln(x))+C