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

The integral of-(sin(ln(x))sin(2ln(x)))/(2x) is -1/4 (sin(ln(x))-1/3 sin(3ln(x)))+C

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

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

- \frac{1}{4} (sin (ln (x)) - \frac{1}{3} sin (3 ln (x))) + C

Solution steps

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

Rewrite using trig identities

= - \frac{1}{2} \cdot \int \frac{- cos ( 3 ln ( x ) ) + cos ( - 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 \frac{1}{2} \cdot \int \frac{- cos ( 3 ln ( x ) ) + cos ( - ln ( x ) )}{x} d x

Simplify

\frac{- cos ( 3 ln ( x ) ) + cos ( - ln ( x ) )}{x} : \frac{cos ( ln ( x ) ) - cos ( 3 ln ( x ) )}{x}

= - \frac{1}{2} \cdot \frac{1}{2} \cdot \int \frac{cos ( ln ( x ) ) - cos ( 3 ln ( x ) )}{x} d x

Expand

\frac{cos ( ln ( x ) ) - cos ( 3 ln ( x ) )}{x} : \frac{cos ( ln ( x ) )}{x} - \frac{cos ( 3 ln ( x ) )}{x}

= - \frac{1}{2} \cdot \frac{1}{2} \cdot \int \frac{cos ( ln ( x ) )}{x} - \frac{cos ( 3 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} \cdot \frac{1}{2} (\int \frac{cos ( ln ( x ) )}{x} d x - \int \frac{cos ( 3 ln ( x ) )}{x} d x)

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

\int \frac{cos ( 3 ln ( x ) )}{x} d x = \frac{1}{3} sin (3 ln (x))

= - \frac{1}{2} \cdot \frac{1}{2} (sin (ln (x)) - \frac{1}{3} sin (3 ln (x)))

Simplify

- \frac{1}{2} \cdot \frac{1}{2} (sin (ln (x)) - \frac{1}{3} sin (3 ln (x))) : - \frac{1}{4} (sin (ln (x)) - \frac{1}{3} sin (3 ln (x)))

= - \frac{1}{4} (sin (ln (x)) - \frac{1}{3} sin (3 ln (x)))

Add a constant to the solution

= - \frac{1}{4} (sin (ln (x)) - \frac{1}{3} sin (3 ln (x))) + C

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What is the integral of-(sin(ln(x))sin(2ln(x)))/(2x) ?
The integral of-(sin(ln(x))sin(2ln(x)))/(2x) is -1/4 (sin(ln(x))-1/3 sin(3ln(x)))+C