What is the integral of (cos(ln(t)))/(3t) ?

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

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integral of (cos(ln(t)))/(3t)

\int \frac{cos ( ln ( t ) )}{3 t} d t

\frac{1}{3} sin (ln (t)) + C

Solution steps

\int \frac{cos ( ln ( t ) )}{3 t} d t

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{cos ( ln ( t ) )}{t} d t

Apply u-substitution

: \int cos (u) d u

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

Use the common integral:

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

= \frac{1}{3} sin (u)

Substitute back

u = ln (t)

= \frac{1}{3} sin (ln (t))

Add a constant to the solution

= \frac{1}{3} sin (ln (t)) + C

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