integral of cot^3(x)sin^7(x)

解答

\int cot^{3} (x) sin^{7} (x) d x

\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7} + C

求解步骤

\int cot^{3} (x) sin^{7} (x) d x

使用三角恒等式改写

= \int (1 - sin^{2} (x)) cos (x) sin^{4} (x) d x

使用换元积分法

= \int u^{4} (1 - u^{2}) d u

乘开

u^{4} (1 - u^{2}) : u^{4} - u^{6}

= \int u^{4} - u^{6} 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^{4} d u - \int u^{6} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= \frac{u ^{5}}{5} - \frac{u ^{7}}{7}

u = sin (x)

代回

= \frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7}

解答补常数

= \frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7} + C

\int \frac{4 x}{2 x + 1} d x

\int x (3 x^{2} - 5)^{7} d x

\int \frac{4 x + 4}{x + 4} d x

\int \frac{5}{3 x ^{2}} d x

\int cos (5 x) sin^{2} (5 x) d x