zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral from 0 to 1 of x^2(3+4x^3)^3

解答

\int_{0}^{1} x^{2} (3 + 4 x^{3})^{3} d x

解答

\frac{145}{3}

+1

十进制

48.33333 \dots

求解步骤

\int_{0}^{1} x^{2} (3 + 4 x^{3})^{3} d x

使用换元积分法

= \int_{0}^{1} \frac{( 3 + 4 u ) ^{3}}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int_{0}^{1} (3 + 4 u)^{3} d u

使用换元积分法

= \frac{1}{3} \cdot \int_{3}^{7} \frac{v ^{3}}{4} d v

提出常数:

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

= \frac{1}{3} \cdot \frac{1}{4} \cdot \int_{3}^{7} v^{3} d v

使用幂法则

= \frac{1}{3} \cdot \frac{1}{4} [\frac{v ^{4}}{4}]_{3}^{7}

化简

\frac{1}{3} \cdot \frac{1}{4} [\frac{v ^{4}}{4}]_{3}^{7} : \frac{1}{12} [\frac{v ^{4}}{4}]_{3}^{7}

= \frac{1}{12} [\frac{v ^{4}}{4}]_{3}^{7}

带入上下限计算后得:

580

= \frac{1}{12} \cdot 580

化简

= \frac{145}{3}

作图

流行的例子

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

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

integral from 0 to 4 of 3e^x

\int_{0}^{4} 3 e^{x} d x

integral from-pi to pi of (1+x)cos(nx)

\int_{- π}^{π} (1 + x) cos (n x) d x

integral from 5 to 10 of 2/((x+2)^2)

\int_{5}^{10} \frac{2}{( x + 2 ) ^{2}} d x

integral from-pi to pi of sin(x)sin(x)

\int_{- π}^{π} sin (x) sin (x) d x