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受欢迎的微积分 >

integral of (e^{3x})/(e^{2x)-e^x-6}

解答

\int \frac{e ^{3 x}}{e ^{2 x} - e ^{x} - 6} d x

解答

e^{x} - \frac{1}{2} - \frac{4}{5} ln ∣ 2 e^{x} + 4 ∣ + \frac{9}{5} ln ∣ 2 e^{x} - 6 ∣ + C

求解步骤

\int \frac{e ^{3 x}}{e ^{2 x} - e ^{x} - 6} d x

使用换元积分法

= \int \frac{u ^{2}}{u ^{2} - u - 6} d u

对

u^{2} - u - 6

配方:

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

= \int \frac{u ^{2}}{( u - \frac{1}{2} ) ^{2} - \frac{25}{4}} d u

使用换元积分法

= \int \frac{( 2 v + 1 ) ^{2}}{4 v ^{2} - 25} d v

将

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

用部份分式展开:

1 - \frac{8}{5 ( 2 v + 5 )} + \frac{18}{5 ( 2 v - 5 )}

= \int 1 - \frac{8}{5 ( 2 v + 5 )} + \frac{18}{5 ( 2 v - 5 )} d v

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

= \int 1 d v - \int \frac{8}{5 ( 2 v + 5 )} d v + \int \frac{18}{5 ( 2 v - 5 )} d v

\int 1 d v = v

\int \frac{8}{5 ( 2 v + 5 )} d v = \frac{4}{5} ln ∣ 2 v + 5 ∣

\int \frac{18}{5 ( 2 v - 5 )} d v = \frac{9}{5} ln ∣ 2 v - 5 ∣

= v - \frac{4}{5} ln ∣ 2 v + 5 ∣ + \frac{9}{5} ln ∣ 2 v - 5 ∣

代回

= e^{x} - \frac{1}{2} - \frac{4}{5} ln 2 (e^{x} - \frac{1}{2}) + 5 + \frac{9}{5} ln 2 (e^{x} - \frac{1}{2}) - 5

化简

e^{x} - \frac{1}{2} - \frac{4}{5} ln 2 (e^{x} - \frac{1}{2}) + 5 + \frac{9}{5} ln 2 (e^{x} - \frac{1}{2}) - 5 : e^{x} - \frac{1}{2} - \frac{4}{5} ln ∣ 2 e^{x} + 4 ∣ + \frac{9}{5} ln ∣ 2 e^{x} - 6 ∣

= e^{x} - \frac{1}{2} - \frac{4}{5} ln ∣ 2 e^{x} + 4 ∣ + \frac{9}{5} ln ∣ 2 e^{x} - 6 ∣

解答补常数

= e^{x} - \frac{1}{2} - \frac{4}{5} ln ∣ 2 e^{x} + 4 ∣ + \frac{9}{5} ln ∣ 2 e^{x} - 6 ∣ + C

作图

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