What is the integral from-pi to pi of (x^2+x)cos(nx) ?

The integral from-pi to pi of (x^2+x)cos(nx) is (4pi(-1)^n)/(n^2)

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integral from-pi to pi of (x^2+x)cos(nx)

\int_{- π}^{π} (x^{2} + x) cos (n x) d x

\frac{4 π ( - 1 ) ^{n}}{n ^{2}}

Solution steps

\int_{- π}^{π} (x^{2} + x) cos (n x) d x

Expand

(x^{2} + x) cos (n x) : x^{2} cos (n x) + x cos (n x)

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

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

\int_{- π}^{π} x cos (n x) d x = 0

= \frac{4 π ( - 1 ) ^{n}}{n ^{2}} + 0

Simplify

= \frac{4 π ( - 1 ) ^{n}}{n ^{2}}

\int_{0}^{2} x^{2} (x^{2} + 1) d x

\int_{3}^{5} (2 x - 5)^{4} d x

\int_{- 2}^{2} x^{2} + 2 d x

\int_{1}^{3} 3 x (x^{2} - 5)^{5} d x

\int_{0}^{3} \frac{1}{3 x} d x

What is the integral from-pi to pi of (x^2+x)cos(nx) ?
The integral from-pi to pi of (x^2+x)cos(nx) is (4pi(-1)^n)/(n^2)