What is the integral of (sin(ln(x)))/(2x) ?

The integral of (sin(ln(x)))/(2x) is -1/2 cos(ln(x))+C

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integral of (sin(ln(x)))/(2x)

\int \frac{sin ( ln ( x ) )}{2 x} d x

- \frac{1}{2} cos (ln (x)) + C

Solution steps

\int \frac{sin ( ln ( x ) )}{2 x} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{sin ( ln ( x ) )}{x} d x

Apply u-substitution

: \int sin (u) d u

= \frac{1}{2} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{2} (- cos (u))

Substitute back

u = ln (x)

= \frac{1}{2} (- cos (ln (x)))

Simplify

= - \frac{1}{2} cos (ln (x))

Add a constant to the solution

= - \frac{1}{2} cos (ln (x)) + C

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\int \frac{16}{1 + 4 x ^{2}} d x

\int 3 x^{- \frac{3}{2}} d x

What is the integral of (sin(ln(x)))/(2x) ?
The integral of (sin(ln(x)))/(2x) is -1/2 cos(ln(x))+C