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

The integral of (ln(x)cos(ln(x)))/x is ln(x)sin(ln(x))+cos(ln(x))+C

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

\int \frac{ln ( x ) cos ( ln ( x ) )}{x} d x

ln (x) sin (ln (x)) + cos (ln (x)) + C

Solution steps

\int \frac{ln ( x ) cos ( ln ( x ) )}{x} d x

Apply u-substitution

: \int u cos (u) d u

= \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= u sin (u) - \int sin (u) d u

\int sin (u) d u = - cos (u)

= u sin (u) - (- cos (u))

Substitute back

u = ln (x)

= ln (x) sin (ln (x)) - (- cos (ln (x)))

Simplify

= ln (x) sin (ln (x)) + cos (ln (x))

Add a constant to the solution

= ln (x) sin (ln (x)) + cos (ln (x)) + C

\int \frac{2 x ^{2} + 5 x + 16}{x ^{4} + 8 x ^{2} + 16} d x

\int \frac{ln ( 2 x )}{ln ( 4 x )} \frac{1}{x} d x

\int sin^{2} (\frac{x}{8}) cos (\frac{x}{8}) d x

\int \frac{x}{49 - x ^{2}} d x

\int \frac{1 - x}{5 - 2 x - x ^{2}} d x

What is the integral of (ln(x)cos(ln(x)))/x ?
The integral of (ln(x)cos(ln(x)))/x is ln(x)sin(ln(x))+cos(ln(x))+C