zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of (3e^{2x})/(e^{2x)+15e^x+54}

解答

\int \frac{3 e ^{2 x}}{e ^{2 x} + 15 e ^{x} + 54} d x

解答

6 (\frac{1}{4} ln 60 e^{x} + 4 e^{2 x} + 216 + \frac{5}{4} (ln \frac{2}{3} e^{x} + 6 - ln \frac{2}{3} e^{x} + 4)) + C

求解步骤

\int \frac{3 e ^{2 x}}{e ^{2 x} + 15 e ^{x} + 54} d x

提出常数:

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

= 3 \cdot \int \frac{e ^{2 x}}{e ^{2 x} + 15 e ^{x} + 54} d x

使用换元积分法

= 3 \cdot \int \frac{u}{u ^{2} + 15 u + 54} d u

对

u^{2} + 15 u + 54

配方:

(u + \frac{15}{2})^{2} - \frac{9}{4}

= 3 \cdot \int \frac{u}{( u + \frac{15}{2} ) ^{2} - \frac{9}{4}} d u

使用换元积分法

= 3 \cdot \int \frac{2 ( 2 v - 15 )}{4 v ^{2} - 9} d v

提出常数:

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

= 3 \cdot 2 \cdot \int \frac{2 v - 15}{4 v ^{2} - 9} d v

乘开

\frac{2 v - 15}{4 v ^{2} - 9} : \frac{2 v}{4 v ^{2} - 9} - \frac{15}{4 v ^{2} - 9}

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

= 3 \cdot 2 (\int \frac{2 v}{4 v ^{2} - 9} d v - \int \frac{15}{4 v ^{2} - 9} d v)

\int \frac{2 v}{4 v ^{2} - 9} d v = \frac{1}{4} ln 4 v^{2} - 9

\int \frac{15}{4 v ^{2} - 9} d v = - \frac{5}{4} (ln \frac{2}{3} v + 1 - ln \frac{2}{3} v - 1)

= 3 \cdot 2 (\frac{1}{4} ln 4 v^{2} - 9 - (- \frac{5}{4} (ln \frac{2}{3} v + 1 - ln \frac{2}{3} v - 1)))

代回

= 3 \cdot 2 (\frac{1}{4} ln 4 (e^{x} + \frac{15}{2})^{2} - 9 - (- \frac{5}{4} (ln \frac{2}{3} (e^{x} + \frac{15}{2}) + 1 - ln \frac{2}{3} (e^{x} + \frac{15}{2}) - 1)))

化简

3 \cdot 2 (\frac{1}{4} ln 4 (e^{x} + \frac{15}{2})^{2} - 9 - (- \frac{5}{4} (ln \frac{2}{3} (e^{x} + \frac{15}{2}) + 1 - ln \frac{2}{3} (e^{x} + \frac{15}{2}) - 1))) : 6 (\frac{1}{4} ln 60 e^{x} + 4 e^{2 x} + 216 + \frac{5}{4} (ln \frac{2}{3} e^{x} + 6 - ln \frac{2}{3} e^{x} + 4))

= 6 (\frac{1}{4} ln 60 e^{x} + 4 e^{2 x} + 216 + \frac{5}{4} (ln \frac{2}{3} e^{x} + 6 - ln \frac{2}{3} e^{x} + 4))

解答补常数

= 6 (\frac{1}{4} ln 60 e^{x} + 4 e^{2 x} + 216 + \frac{5}{4} (ln \frac{2}{3} e^{x} + 6 - ln \frac{2}{3} e^{x} + 4)) + C

作图

流行的例子

f (x) = \frac{4}{x ^{2}}

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

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

integral from 0 to infinity of x^3e^{-x}

\int_{0}^{\infty} x^{3} e^{- x} d x

derivative of sech(x)

\frac{d}{d x} (sech (x))

derivative x^3*e^x

d er i v a t i v e x^{3} \cdot e^{x}