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受欢迎的微积分 >

integral from-1 to 2 of (5y)/(y^2-y-6)

解答

\int_{- 1}^{2} \frac{5 y}{y ^{2} - y - 6} d y

解答

- 2 ln (2)

+1

十进制

- 1.38629 \dots

求解步骤

\int_{- 1}^{2} \frac{5 y}{y ^{2} - y - 6} d y

提出常数:

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

= 5 \cdot \int_{- 1}^{2} \frac{y}{y ^{2} - y - 6} d y

对

y^{2} - y - 6

配方:

(y - \frac{1}{2})^{2} - \frac{25}{4}

= 5 \cdot \int_{- 1}^{2} \frac{y}{( y - \frac{1}{2} ) ^{2} - \frac{25}{4}} d y

使用换元积分法

= 5 \cdot \int_{- \frac{3}{2}}^{\frac{3}{2}} \frac{2 ( 2 u + 1 )}{4 u ^{2} - 25} d u

提出常数:

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

= 5 \cdot 2 \cdot \int_{- \frac{3}{2}}^{\frac{3}{2}} \frac{2 u + 1}{4 u ^{2} - 25} d u

乘开

\frac{2 u + 1}{4 u ^{2} - 25} : \frac{2 u}{4 u ^{2} - 25} + \frac{1}{4 u ^{2} - 25}

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

= 5 \cdot 2 (\int_{- \frac{3}{2}}^{\frac{3}{2}} \frac{2 u}{4 u ^{2} - 25} d u + \int_{- \frac{3}{2}}^{\frac{3}{2}} \frac{1}{4 u ^{2} - 25} d u)

\int_{- \frac{3}{2}}^{\frac{3}{2}} \frac{2 u}{4 u ^{2} - 25} d u = 0

\int_{- \frac{3}{2}}^{\frac{3}{2}} \frac{1}{4 u ^{2} - 25} d u = - \frac{1}{5} ln (2)

= 5 \cdot 2 (0 - \frac{1}{5} ln (2))

化简

5 \cdot 2 (0 - \frac{1}{5} ln (2)) : - 2 ln (2)

= - 2 ln (2)

作图

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