integral from-pi to pi of cos(x)cos(nx)

解答

\int_{- π}^{π} cos (x) cos (n x) d x

0

求解步骤

\int_{- π}^{π} cos (x) cos (n x) d x

使用三角恒等式改写

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

提出常数:

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

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

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

\int_{- π}^{π} cos (x + n x) d x = 0

\int_{- π}^{π} cos (x - n x) d x = 0

= \frac{1}{2} (0 + 0)

化简

= 0

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