integral from 4 to 8 of y/(y^2-2y-3)

解

\int_{4}^{8} \frac{y}{y ^{2} - 2 y - 3} d y

\frac{2 ln ( 3 ) + 2 ln ( 5 )}{4}

十進法表記

1.35402 \dots

解答ステップ

\int_{4}^{8} \frac{y}{y ^{2} - 2 y - 3} d y

以下の部分分数を得る:

\frac{y}{y ^{2} - 2 y - 3} : \frac{1}{4 ( y + 1 )} + \frac{3}{4 ( y - 3 )}

= \int_{4}^{8} \frac{1}{4 ( y + 1 )} + \frac{3}{4 ( y - 3 )} d y

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{4}^{8} \frac{1}{4 ( y + 1 )} d y + \int_{4}^{8} \frac{3}{4 ( y - 3 )} d y

\int_{4}^{8} \frac{1}{4 ( y + 1 )} d y = \frac{1}{4} (2 ln (3) - ln (5))

\int_{4}^{8} \frac{3}{4 ( y - 3 )} d y = \frac{3}{4} ln (5)

= \frac{1}{4} (2 ln (3) - ln (5)) + \frac{3}{4} ln (5)

簡素化

\frac{1}{4} (2 ln (3) - ln (5)) + \frac{3}{4} ln (5) : \frac{2 ln ( 3 ) + 2 ln ( 5 )}{4}

= \frac{2 ln ( 3 ) + 2 ln ( 5 )}{4}