integral of cos^3(θ)sqrt(sin(θ))

解答

\int cos^{3} (θ) sin (θ) d θ

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

求解步骤

\int cos^{3} (θ) sin (θ) d θ

使用三角恒等式改写

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

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 (θ)

代回

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

解答补常数

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

\int 7 x^{6} (x^{7} + 2)^{5} d x

\int (sin (x))^{6} (cos (x))^{5} d x

\int \frac{8}{9 x 81 x ^{2} - 64} d x

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

\int \frac{x ^{2} + 3 x + 2}{x ^{3} + 4 x} d x