integral from 0 to pi/3 of cos^3(x)

解答

\int_{0}^{\frac{π}{3}} cos^{3} (x) d x

\frac{3 3}{8}

十进制

0.64951 \dots

求解步骤

\int_{0}^{\frac{π}{3}} cos^{3} (x) d x

使用三角恒等式改写

= \int_{0}^{\frac{π}{3}} (1 - sin^{2} (x)) cos (x) d x

使用换元积分法

= \int_{0}^{\frac{3}{2}} 1 - u^{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_{0}^{\frac{3}{2}} 1 d u - \int_{0}^{\frac{3}{2}} u^{2} d u

\int_{0}^{\frac{3}{2}} 1 d u = \frac{3}{2}

\int_{0}^{\frac{3}{2}} u^{2} d u = \frac{3}{8}

= \frac{3}{2} - \frac{3}{8}

化简

\frac{3}{2} - \frac{3}{8} : \frac{3 3}{8}

= \frac{3 3}{8}

(x y^{2} + x^{3}) d x + (y^{3} + x^{2} y) d y = 0

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

\int (4 - 3 sin (2 x))^{4} cos (2 x) d x

2 y y^{^{'}} + 2 = y^{2} + 2 x

\frac{d}{d x} (\frac{x ^{2}}{4})