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受欢迎的微积分 >

integral from 0 to 1 of 3x^2e^{x^3}

解答

\int_{0}^{1} 3 x^{2} e^{x^{3}} d x

解答

e - 1

+1

十进制

1.71828 \dots

求解步骤

\int_{0}^{1} 3 x^{2} e^{x^{3}} d x

提出常数:

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

= 3 \cdot \int_{0}^{1} x^{2} e^{x^{3}} d x

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

\int e^{u} d u = e^{u}

= 3 \cdot \frac{1}{3} [e^{u}]_{0}^{1}

化简

3 \cdot \frac{1}{3} [e^{u}]_{0}^{1} : [e^{u}]_{0}^{1}

= [e^{u}]_{0}^{1}

带入上下限计算后得:

e - 1

= e - 1

作图

流行的例子

integral from 1 to 2 of xsqrt((x-1))

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

integral from-1 to 3 of 2x^3

\int_{- 1}^{3} 2 x^{3} d x

integral from 0 to a of (3x^2-x+1)

\int_{0}^{a} (3 x^{2} - x + 1) d x

integral from 0 to 0.3 of 3e^{-x^2}

\int_{0}^{0.3} 3 e^{- x^{2}} d x

integral from 0 to 2 of xsqrt(2x^2+1)

\int_{0}^{2} x 2 x^{2} + 1 d x