integral from 0 to pi/6 of cos^3(x)

解答

\int_{0}^{\frac{π}{6}} cos^{3} (x) d x

\frac{11}{24}

十进制

0.45833 \dots

求解步骤

\int_{0}^{\frac{π}{6}} cos^{3} (x) d x

使用三角恒等式改写

= \int_{0}^{\frac{π}{6}} (1 - sin^{2} (x)) cos (x) d x

使用换元积分法

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

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

\int_{0}^{\frac{1}{2}} u^{2} d u = \frac{1}{24}

= \frac{1}{2} - \frac{1}{24}

化简

\frac{1}{2} - \frac{1}{24} : \frac{11}{24}

= \frac{11}{24}

\int 16 x ln (7 x) d x

\int_{0}^{\frac{π}{2}} 11 sin^{2} (2 x) d x

\frac{d}{d x} (x^{3} \cdot sin^{2} (5 x))

\frac{\partial}{\partial x} (\frac{sin ( x )}{sin ( y )})

\int 3 x - x^{2} d x