What is the integral of (sin(piln(x)))/x ?

The integral of (sin(piln(x)))/x is -1/pi cos(piln(x))+C

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integral of (sin(piln(x)))/x

\int \frac{sin ( π ln ( x ) )}{x} d x

- \frac{1}{π} cos (π ln (x)) + C

Solution steps

\int \frac{sin ( π ln ( x ) )}{x} d x

Apply u-substitution

: \int sin (π u) d u

= \int sin (π u) d u

Apply u-substitution

: \int sin (v) \frac{1}{π} d v

= \int sin (v) \frac{1}{π} d v

Take the constant out:

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

= \frac{1}{π} \cdot \int sin (v) d v

Use the common integral:

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

= \frac{1}{π} (- cos (v))

Substitute back

= \frac{1}{π} (- cos (π ln (x)))

Simplify

= - \frac{1}{π} cos (π ln (x))

Add a constant to the solution

= - \frac{1}{π} cos (π ln (x)) + C

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