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

The integral of ln(x)*cos(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 ln(x)*cos(ln(x))

\int ln (x) \cdot cos (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 ln (x) cos (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 ln(x)*cos(ln(x)) ?
The integral of ln(x)*cos(ln(x)) is ln(x)(1/2 xsin(ln(x))+1/2 xcos(ln(x)))-1/2 xsin(ln(x))+C