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

The integral of-sin(x)ln(x) is ln(x)cos(x)-\Ci(x)+C

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integral of-sin(x)ln(x)

\int - sin (x) ln (x) d x

ln (x) cos (x) - Ci (x) + C

Solution steps

\int - sin (x) ln (x) d x

Take the constant out:

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

= - \int sin (x) ln (x) d x

Apply Integration By Parts

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

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

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

= - (- ln (x) cos (x) - (- Ci (x)))

Simplify

= ln (x) cos (x) - Ci (x)

Add a constant to the solution

= ln (x) cos (x) - Ci (x) + C

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What is the integral of-sin(x)ln(x) ?
The integral of-sin(x)ln(x) is ln(x)cos(x)-\Ci(x)+C