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

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

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integral of cos(ln(x))ln(x)

\int cos (ln (x)) ln (x) d x

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

Solution steps

\int cos (ln (x)) ln (x) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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