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受欢迎的微积分 >

integral from 0 to 1 of 5(1-x)^4

解答

\int_{0}^{1} 5 (1 - x)^{4} d x

解答

1

求解步骤

\int_{0}^{1} 5 (1 - x)^{4} d x

提出常数:

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

= 5 \cdot \int_{0}^{1} (1 - x)^{4} d x

使用换元积分法

= 5 \cdot \int_{1}^{0} - u^{4} d u

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

= 5 (- \int_{0}^{1} - u^{4} d u)

提出常数:

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

= 5 (- (- \int_{0}^{1} u^{4} d u))

使用幂法则

= 5 (- (- [\frac{u ^{5}}{5}]_{0}^{1}))

化简

= 5 [\frac{u ^{5}}{5}]_{0}^{1}

带入上下限计算后得:

\frac{1}{5}

= 5 \cdot \frac{1}{5}

化简

= 1

作图

流行的例子

integral from-1 to 9 of x^{1/5}-x

\int_{- 1}^{9} x^{\frac{1}{5}} - x d x

integral from 1 to 3 of 8/x

\int_{1}^{3} \frac{8}{x} d x

integral from 10 to 12 of sin(5x)

\int_{10}^{12} sin (5 x) d x

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\int_{1}^{7} \frac{2}{4 x - 1} d x

integral from 1 to cos(x) of 10t^9

\int_{1}^{c o s (x)} 10 t^{9} d t