integral of cos^2((npix)/L)

解答

\int cos^{2} (\frac{nπ x}{L}) d x

\frac{1}{2} (x + \frac{L}{2 πn} sin (\frac{2 πn x}{L})) + C

求解步骤

\int cos^{2} (\frac{nπ x}{L}) d x

使用三角恒等式改写

= \int \frac{1}{2} (1 + cos (\frac{2 πn x}{L})) 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 1 + cos (\frac{2 πn x}{L}) 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 1 d x + \int cos (\frac{2 πn x}{L}) d x)

\int 1 d x = x

\int cos (\frac{2 πn x}{L}) d x = \frac{L}{2 πn} sin (\frac{2 πn x}{L})

= \frac{1}{2} (x + \frac{L}{2 πn} sin (\frac{2 πn x}{L}))

解答补常数

= \frac{1}{2} (x + \frac{L}{2 πn} sin (\frac{2 πn x}{L})) + C

\int 4 x^{3} + 3 x^{4} d x

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\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = \frac{e ^{x}}{1 + x ^{2}}

n \to \infty lim ({(1 + \frac{2}{n})^{4 n}})