integral from 0 to 1 of 2y^2(1-y)^{x-1}

解答

\int_{0}^{1} 2 y^{2} (1 - y)^{x - 1} d y

2 (\frac{1}{x + 2} - \frac{2}{x + 1} + \frac{1}{x})

求解步骤

\int_{0}^{1} 2 y^{2} (1 - y)^{x - 1} d y

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{1} (1 - y)^{- 1 + x} y^{2} d y

使用换元积分法

= 2 \cdot \int_{1}^{0} - (- u + 1)^{2} u^{- 1 + x} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 2 (- \int_{0}^{1} - (- u + 1)^{2} u^{- 1 + x} d u)

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 (- (- \int_{0}^{1} (- u + 1)^{2} u^{- 1 + x} d u))

乘开

(- u + 1)^{2} u^{- 1 + x} : u^{x + 1} - 2 u^{x} + u^{- 1 + x}

= 2 (- (- \int_{0}^{1} u^{x + 1} - 2 u^{x} + u^{- 1 + x} 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

= 2 (- (- (\int_{0}^{1} u^{x + 1} d u - \int_{0}^{1} 2 u^{x} d u + \int_{0}^{1} u^{- 1 + x} d u)))

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

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

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

= 2 (- (- (\frac{1}{x + 2} - \frac{2}{x + 1} + \frac{1}{x})))

化简

= 2 (\frac{1}{x + 2} - \frac{2}{x + 1} + \frac{1}{x})

\int_{- 2}^{1} (x - 2)^{2} d x

\int_{6}^{9} (6 - x) (x - 9) d x

\int_{- 2}^{6} ∣ x ∣ d x

\int_{0}^{1} e^{- z^{2}} d z

\int_{0}^{100} 5 a x d x