What is the integral from 0 to pi of sin(x)sin(n)x ?

The integral from 0 to pi of sin(x)sin(n)x is sin(n)pi

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integral from 0 to pi of sin(x)sin(n)x

\int_{0}^{π} sin (x) sin (n) x d x

sin (n) π

Solution steps

\int_{0}^{π} sin (x) sin (n) x d x

Take the constant out:

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

= sin (n) \cdot \int_{0}^{π} sin (x) x d x

Apply Integration By Parts

= sin (n) [- x cos (x) - \int - cos (x) d x]_{0}^{π}

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

= sin (n) [- x cos (x) - (- sin (x))]_{0}^{π}

Simplify

= sin (n) [- x cos (x) + sin (x)]_{0}^{π}

Compute the boundaries:

π

= sin (n) π

What is the integral from 0 to pi of sin(x)sin(n)x ?
The integral from 0 to pi of sin(x)sin(n)x is sin(n)pi