What is the integral of (cos(ln(x)))/(5x) ?

The integral of (cos(ln(x)))/(5x) is 1/5 sin(ln(x))+C

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integral of (cos(ln(x)))/(5x)

\int \frac{cos ( ln ( x ) )}{5 x} d x

\frac{1}{5} sin (ln (x)) + C

Solution steps

\int \frac{cos ( ln ( x ) )}{5 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int \frac{cos ( ln ( x ) )}{x} d x

Apply u-substitution

: \int cos (u) d u

= \frac{1}{5} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{5} sin (u)

Substitute back

u = ln (x)

= \frac{1}{5} sin (ln (x))

Add a constant to the solution

= \frac{1}{5} sin (ln (x)) + C

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What is the integral of (cos(ln(x)))/(5x) ?
The integral of (cos(ln(x)))/(5x) is 1/5 sin(ln(x))+C