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受欢迎的微积分 >

integral of (e^t)/(e^{2t)+3e^t+2}

解答

\int \frac{e ^{t}}{e ^{2 t} + 3 e ^{t} + 2} d t

解答

- ln 2 e^{t} + 4 + ln 2 e^{t} + 2 + C

求解步骤

\int \frac{e ^{t}}{e ^{2 t} + 3 e ^{t} + 2} d t

使用换元积分法

= \int \frac{1}{u ^{2} + 3 u + 2} d u

对

u^{2} + 3 u + 2

配方:

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

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

使用换元积分法

= \int \frac{4}{4 v ^{2} - 1} d v

提出常数:

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

= 4 \cdot \int \frac{1}{4 v ^{2} - 1} d v

因式分解

4 v^{2} - 1 : - (- 4 v^{2} + 1)

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

= 4 (- \frac{1}{2} \cdot \int \frac{1}{- w ^{2} + 1} d w)

使用积分常见定则:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= 4 (- \frac{1}{2} (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代回

= 4 (- \frac{1}{2} (\frac{ln 2 ( e ^{t} + \frac{3}{2} ) + 1}{2} - \frac{ln 2 ( e ^{t} + \frac{3}{2} ) - 1}{2}))

化简

4 (- \frac{1}{2} (\frac{ln 2 ( e ^{t} + \frac{3}{2} ) + 1}{2} - \frac{ln 2 ( e ^{t} + \frac{3}{2} ) - 1}{2})) : - ln 2 e^{t} + 4 + ln 2 e^{t} + 2

= - ln 2 e^{t} + 4 + ln 2 e^{t} + 2

解答补常数

= - ln 2 e^{t} + 4 + ln 2 e^{t} + 2 + C

作图

流行的例子

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\int_{y}^{2} \frac{3}{16} x y d x

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x \to 1 lim (\frac{4 x - 4}{x - 1})

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\int 16 x sin (x) d x

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t a y l or \frac{1 + x}{1 - x}