integral from 0 to a of sin^2((3pix)/a)

解答

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

\frac{1}{2} a

求解步骤

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

使用三角恒等式改写

= \int_{0}^{a} \frac{1}{2} (1 - cos (\frac{6 π x}{a})) 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}^{a} 1 - cos (\frac{6 π x}{a}) 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}^{a} 1 d x - \int_{0}^{a} cos (\frac{6 π x}{a}) d x)

\int_{0}^{a} 1 d x = a

\int_{0}^{a} cos (\frac{6 π x}{a}) d x = 0

= \frac{1}{2} (a - 0)

化简

= \frac{1}{2} a

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