integral from 0 to L of sin^2((npix)/L)

פתרון

\int_{0}^{L} sin^{2} (\frac{nπ x}{L}) d x

\frac{1}{2} (L - \frac{L sin ( 2 πn )}{2 πn})

צעדי פתרון

\int_{0}^{L} sin^{2} (\frac{nπ x}{L}) d x

Rewrite using trig identities

= \int_{0}^{L} \frac{1}{2} (1 - cos (\frac{2 πn x}{L})) d x

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

:הפעל את תכונת הליניאריות

= \frac{1}{2} \cdot \int_{0}^{L} 1 - cos (\frac{2 πn x}{L}) d x

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

:הפעל את תכונת הליניאריות

= \frac{1}{2} (\int_{0}^{L} 1 d x - \int_{0}^{L} cos (\frac{2 πn x}{L}) d x)

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

\int_{0}^{L} cos (\frac{2 πn x}{L}) d x = \frac{L sin ( 2 πn )}{2 πn}

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

d er i v a t i v e y = \frac{e ^{x}}{1 + x}

d er i v a t i v e tan^{2} (6 x)

\int_{1}^{3} \frac{x ^{2} + 3}{4 x - x ^{2}} d x

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x^{4} + y^{3} x^{5} + 4 \frac{d y}{d x} = 0, y (1) = 0