What is the integral from-pi to pi of x^2e^{-inx} ?

The integral from-pi to pi of x^2e^{-inx} is (4pi(-1)^n)/(n^2)

Popular >

integral from-pi to pi of x^2e^{-inx}

\int_{- π}^{π} x^{2} e^{- in x} d x

\frac{4 π ( - 1 ) ^{n}}{n ^{2}}

Solution steps

\int_{- π}^{π} x^{2} e^{- in x} d x

Simplify

x^{2} e^{- in x} : x^{2} cos (n x) - i x^{2} sin (n x)

= \int_{- π}^{π} x^{2} cos (n x) - i x^{2} 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^{2} cos (n x) d x - \int_{- π}^{π} i x^{2} sin (n x) d x

\int_{- π}^{π} x^{2} cos (n x) d x = \frac{4 π ( - 1 ) ^{n}}{n ^{2}}

\int_{- π}^{π} i x^{2} sin (n x) d x = 0

= \frac{4 π ( - 1 ) ^{n}}{n ^{2}} - 0

Simplify

= \frac{4 π ( - 1 ) ^{n}}{n ^{2}}

\int_{3}^{4} 6 x - 24 d x

\int_{- 1}^{0} - x sin (nπ x) d x

\int_{1}^{3} x^{2} - 2 x - 3 d x

\int_{2}^{B} (2 x - 3)^{4} d x

\int_{0}^{\frac{3 π}{2}} cos (2 x) d x

What is the integral from-pi to pi of x^2e^{-inx} ?
The integral from-pi to pi of x^2e^{-inx} is (4pi(-1)^n)/(n^2)