integral of sin^{1/2}(x)cos^3(x)

解答

\int sin^{\frac{1}{2}} (x) cos^{3} (x) d x

\frac{2}{3} sin^{\frac{3}{2}} (x) - \frac{2}{7} sin^{\frac{7}{2}} (x) + C

求解步骤

\int sin (x) cos^{3} (x) d x

使用三角恒等式改写

= \int (1 - sin^{2} (x)) cos (x) sin (x) d x

使用换元积分法

= \int u (1 - u^{2}) d u

乘开

u (1 - u^{2}) : u - u^{\frac{5}{2}}

= \int u - u^{\frac{5}{2}} d u

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

= \int u d u - \int u^{\frac{5}{2}} d u

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

= \frac{2}{3} u^{\frac{3}{2}} - \frac{2}{7} u^{\frac{7}{2}}

u = sin (x)

代回

= \frac{2}{3} sin^{\frac{3}{2}} (x) - \frac{2}{7} sin^{\frac{7}{2}} (x)

解答补常数

= \frac{2}{3} sin^{\frac{3}{2}} (x) - \frac{2}{7} sin^{\frac{7}{2}} (x) + C

n \to \infty lim (i = 1 \sum n \frac{i ^{3}}{n ^{4}} + \frac{1}{n})

\frac{d}{d x} (x cos (y) - y e^{x})

x \to 5 lim (\frac{x ^{2} - 7 x + 10}{x ^{2} - 25})

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

\int \frac{sec ^{6} ( x )}{tan ^{3} ( x )} d x