What is the integral of (sin(3ln(x)))/x ?

The integral of (sin(3ln(x)))/x is -1/3 cos(3ln(x))+C

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integral of (sin(3ln(x)))/x

\int \frac{sin ( 3 ln ( x ) )}{x} d x

- \frac{1}{3} cos (3 ln (x)) + C

Solution steps

\int \frac{sin ( 3 ln ( x ) )}{x} d x

Apply u-substitution

= \int \frac{sin ( u )}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{3} (- cos (u))

Substitute back

u = 3 ln (x)

= \frac{1}{3} (- cos (3 ln (x)))

Simplify

= - \frac{1}{3} cos (3 ln (x))

Add a constant to the solution

= - \frac{1}{3} cos (3 ln (x)) + C

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