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

解答

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

\frac{4 π ( - 1 ) ^{n}}{n ^{2}}

求解步骤

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

乘开

(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

使用积分加法定则:

\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

化简

= \frac{4 π ( - 1 ) ^{n}}{n ^{2}}

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