What is the integral of (sin(9ln(t)))/t ?

The integral of (sin(9ln(t)))/t is -1/9 cos(9ln(t))+C

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integral of (sin(9ln(t)))/t

\int \frac{sin ( 9 ln ( t ) )}{t} d t

- \frac{1}{9} cos (9 ln (t)) + C

Solution steps

\int \frac{sin ( 9 ln ( t ) )}{t} d t

Apply u-substitution

= \int \frac{sin ( u )}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{9} (- cos (u))

Substitute back

u = 9 ln (t)

= \frac{1}{9} (- cos (9 ln (t)))

Simplify

= - \frac{1}{9} cos (9 ln (t))

Add a constant to the solution

= - \frac{1}{9} cos (9 ln (t)) + C

What is the integral of (sin(9ln(t)))/t ?
The integral of (sin(9ln(t)))/t is -1/9 cos(9ln(t))+C