integral of+sin^3(x)cos^{14}(x)

解答

\int + sin^{3} (x) cos^{14} (x) d x

- \frac{cos ^{15} ( x )}{15} + \frac{cos ^{17} ( x )}{17} + C

求解步骤

\int sin^{3} (x) cos^{14} (x) d x

使用三角恒等式改写

= \int (1 - cos^{2} (x)) sin (x) cos^{14} (x) d x

使用换元积分法

= \int - u^{14} (1 - u^{2}) d u

乘开

- u^{14} (1 - u^{2}) : - u^{14} + u^{16}

= \int - u^{14} + u^{16} 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^{14} d u + \int u^{16} d u

\int u^{14} d u = \frac{u ^{15}}{15}

\int u^{16} d u = \frac{u ^{17}}{17}

= - \frac{u ^{15}}{15} + \frac{u ^{17}}{17}

u = cos (x)

代回

= - \frac{cos ^{15} ( x )}{15} + \frac{cos ^{17} ( x )}{17}

解答补常数

= - \frac{cos ^{15} ( x )}{15} + \frac{cos ^{17} ( x )}{17} + C

d er i v a t i v e \frac{3 x}{x - \frac{3}{x}}

d er i v a t i v e \frac{35}{4 + x + x}

\frac{d}{d x} (\frac{x - 6}{x + 4})

x \to \infty lim (4 (x - 3))

x \to \infty lim (x (0))