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人気のある微分積分 >

integral from 2 to 3 of 6/(2x^2+2x-4)

解

\int_{2}^{3} \frac{6}{2 x ^{2} + 2 x - 4} d x

解

3 ln (2) - ln (5)

+1

十進法表記

0.47000 \dots

解答ステップ

\int_{2}^{3} \frac{6}{2 x ^{2} + 2 x - 4} d x

定数を除く:

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

= 6 \cdot \int_{2}^{3} \frac{1}{2 x ^{2} + 2 x - 4} d x

以下の部分分数を得る:

\frac{1}{2 x ^{2} + 2 x - 4} : \frac{1}{6 ( x - 1 )} - \frac{1}{6 ( x + 2 )}

= 6 \cdot \int_{2}^{3} \frac{1}{6 ( x - 1 )} - \frac{1}{6 ( x + 2 )} d x

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

= 6 (\int_{2}^{3} \frac{1}{6 ( x - 1 )} d x - \int_{2}^{3} \frac{1}{6 ( x + 2 )} d x)

\int_{2}^{3} \frac{1}{6 ( x - 1 )} d x = \frac{1}{6} ln (2)

\int_{2}^{3} \frac{1}{6 ( x + 2 )} d x = \frac{1}{6} (ln (5) - 2 ln (2))

= 6 (\frac{1}{6} ln (2) - \frac{1}{6} (ln (5) - 2 ln (2)))

簡素化

6 (\frac{1}{6} ln (2) - \frac{1}{6} (ln (5) - 2 ln (2))) : 3 ln (2) - ln (5)

= 3 ln (2) - ln (5)

グラフ

人気の例

integral from 0 to 1 of 3x^2sqrt(x^3-5)

\int_{0}^{1} 3 x^{2} x^{3} - 5 d x

integral from 1 to 5 of (x-1)/(x^2(x+1))

\int_{1}^{5} \frac{x - 1}{x ^{2} ( x + 1 )} d x

integral from 0 to 2 of 5x^2-11x

\int_{0}^{2} 5 x^{2} - 11 x d x

integral from 1 to e of (22+ln(x))^2

\int_{1}^{e} (22 + ln (x))^{2} d x

integral from 0 to 1000 of e^{4z}-5

\int_{0}^{1000} e^{4 z} - 5 d z