What is the integral from-pi to pi of (x+pi)sin(nx) ?

The integral from-pi to pi of (x+pi)sin(nx) is -(2pi(-1)^n)/n

Popular >

integral from-pi to pi of (x+pi)sin(nx)

\int_{- π}^{π} (x + π) sin (n x) d x

- \frac{2 π ( - 1 ) ^{n}}{n}

Solution steps

\int_{- π}^{π} (x + π) sin (n x) d x

Expand

(x + π) sin (n x) : x sin (n x) + π sin (n x)

= \int_{- π}^{π} x sin (n x) + π sin (n 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

= \int_{- π}^{π} x sin (n x) d x + \int_{- π}^{π} π sin (n x) d x

\int_{- π}^{π} x sin (n x) d x = - \frac{2 π ( - 1 ) ^{n}}{n}

\int_{- π}^{π} π sin (n x) d x = 0

= - \frac{2 π ( - 1 ) ^{n}}{n} + 0

Simplify

= - \frac{2 π ( - 1 ) ^{n}}{n}

\int_{0}^{1} \frac{cos ( x )}{x} d x

\int_{0}^{\infty} c e^{- x} d x

\int_{0}^{25} \frac{1}{x} d x

\int_{1}^{3} \frac{9}{x ^{3}} d x

\int_{0}^{1} arctan (d) x d x

What is the integral from-pi to pi of (x+pi)sin(nx) ?
The integral from-pi to pi of (x+pi)sin(nx) is -(2pi(-1)^n)/n