integral of cos^{10}(x)sin^3(x)

解答

\int cos^{10} (x) sin^{3} (x) d x

- \frac{cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13} + C

求解步骤

\int cos^{10} (x) sin^{3} (x) d x

使用三角恒等式改写

= \int (1 - cos^{2} (x)) sin (x) cos^{10} (x) d x

使用换元积分法

= \int - u^{10} (1 - u^{2}) d u

乘开

- u^{10} (1 - u^{2}) : - u^{10} + u^{12}

= \int - u^{10} + u^{12} 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^{10} d u + \int u^{12} d u

\int u^{10} d u = \frac{u ^{11}}{11}

\int u^{12} d u = \frac{u ^{13}}{13}

= - \frac{u ^{11}}{11} + \frac{u ^{13}}{13}

u = cos (x)

代回

= - \frac{cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13}

解答补常数

= - \frac{cos ^{11} ( x )}{11} + \frac{cos ^{13} ( x )}{13} + C

\frac{\partial}{\partial y} (y^{2} - x^{2})

\int (3 + x)^{2} ln (4 x) d x

\frac{d}{d x} ((2 x - e^{2 x})^{s i n (2 x)})

\int lo g_{2} (x) d x

\frac{\partial}{\partial x} (\frac{F ( x , y )}{V ( x , y )} x - \frac{F ( x , y )}{V ( x , y )} y - k y^{n})