What is the integral of (1+sin(ln(x)))/(2x) ?

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

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integral of (1+sin(ln(x)))/(2x)

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

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

Solution steps

\int \frac{1 + 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{1 + sin ( ln ( x ) )}{x} d x

Expand

\frac{1 + sin ( ln ( x ) )}{x} : \frac{1}{x} + \frac{sin ( ln ( x ) )}{x}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{sin ( ln ( x ) )}{x} d x = - cos (ln (x))

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

Add a constant to the solution

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

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