integral from 0 to pi of cos(x)cos(nx)

解答

\int_{0}^{π} cos (x) cos (n x) d x

0

求解步骤

\int_{0}^{π} cos (x) cos (n x) d x

使用三角恒等式改写

= \int_{0}^{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_{0}^{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_{0}^{π} cos (x + n x) d x + \int_{0}^{π} cos (x - n x) d x)

\int_{0}^{π} cos (x + n x) d x = 0

\int_{0}^{π} cos (x - n x) d x = 0

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

化简

= 0

(x^{3} + e^{y}) \frac{d y}{d x} = 3 x^{2}

x \to \infty lim (x - 1)

\frac{d ^{2} y}{d t ^{2}} + 2 \frac{d y}{d t} + y = 0

\frac{d}{d x} (\frac{x}{( x - 9 ) ^{2}})

\frac{d}{d x} (\frac{8 x ^{2} + 9}{x ^{2} + 1})