integral from 3 to 4 of (3x^2)/(x-2)

解答

\int_{3}^{4} \frac{3 x ^{2}}{x - 2} d x

3 (4 ln (2) + \frac{11}{2})

十进制

24.81776 \dots

求解步骤

\int_{3}^{4} \frac{3 x ^{2}}{x - 2} d x

提出常数:

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

= 3 \cdot \int_{3}^{4} \frac{x ^{2}}{x - 2} d x

使用换元积分法

= 3 \cdot \int_{1}^{2} \frac{( u + 2 ) ^{2}}{u} d u

乘开

\frac{( u + 2 ) ^{2}}{u} : u + 4 + \frac{4}{u}

= 3 \cdot \int_{1}^{2} u + 4 + \frac{4}{u} 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

= 3 (\int_{1}^{2} u d u + \int_{1}^{2} 4 d u + \int_{1}^{2} \frac{4}{u} d u)

\int_{1}^{2} u d u = \frac{3}{2}

\int_{1}^{2} 4 d u = 4

\int_{1}^{2} \frac{4}{u} d u = 4 ln (2)

= 3 (\frac{3}{2} + 4 + 4 ln (2))

合并分式

4 + \frac{3}{2} : \frac{11}{2}

= 3 (4 ln (2) + \frac{11}{2})

\int_{0}^{20} q^{2} - 13 q + 70 d q

\int_{4}^{1} x x^{2} + 4 d x

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

\int_{1}^{32} \frac{6}{5} x^{\frac{1}{5}} d x

\int_{0}^{4} (x^{2} - x) d x