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)+13e^x+40}

解答

\int \frac{3 e ^{2 x}}{e ^{2 x} + 13 e ^{x} + 40} d x

解答

6 (\frac{1}{4} ln 52 e^{x} + 4 e^{2 x} + 160 + \frac{13}{12} (ln \frac{16}{3} + \frac{2}{3} e^{x} - ln \frac{10}{3} + \frac{2}{3} e^{x})) + C

求解步骤

\int \frac{3 e ^{2 x}}{e ^{2 x} + 13 e ^{x} + 40} 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} + 13 e ^{x} + 40} d x

使用换元积分法

= 3 \cdot \int \frac{u}{u ^{2} + 13 u + 40} d u

对

u^{2} + 13 u + 40

配方:

(u + \frac{13}{2})^{2} - \frac{9}{4}

= 3 \cdot \int \frac{u}{( u + \frac{13}{2} ) ^{2} - \frac{9}{4}} d u

使用换元积分法

= 3 \cdot \int \frac{2 ( 2 v - 13 )}{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 - 13}{4 v ^{2} - 9} d v

乘开

\frac{2 v - 13}{4 v ^{2} - 9} : \frac{2 v}{4 v ^{2} - 9} - \frac{13}{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{13}{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{13}{4 v ^{2} - 9} d v = - \frac{13}{12} (ln \frac{2}{3} v + 1 - ln \frac{2}{3} v - 1)

= 3 \cdot 2 (\frac{1}{4} ln 4 v^{2} - 9 - (- \frac{13}{12} (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{13}{2})^{2} - 9 - (- \frac{13}{12} (ln \frac{2}{3} (e^{x} + \frac{13}{2}) + 1 - ln \frac{2}{3} (e^{x} + \frac{13}{2}) - 1)))

化简

3 \cdot 2 (\frac{1}{4} ln 4 (e^{x} + \frac{13}{2})^{2} - 9 - (- \frac{13}{12} (ln \frac{2}{3} (e^{x} + \frac{13}{2}) + 1 - ln \frac{2}{3} (e^{x} + \frac{13}{2}) - 1))) : 6 (\frac{1}{4} ln 52 e^{x} + 4 e^{2 x} + 160 + \frac{13}{12} (ln \frac{16}{3} + \frac{2}{3} e^{x} - ln \frac{10}{3} + \frac{2}{3} e^{x}))

= 6 (\frac{1}{4} ln 52 e^{x} + 4 e^{2 x} + 160 + \frac{13}{12} (ln \frac{16}{3} + \frac{2}{3} e^{x} - ln \frac{10}{3} + \frac{2}{3} e^{x}))

解答补常数

= 6 (\frac{1}{4} ln 52 e^{x} + 4 e^{2 x} + 160 + \frac{13}{12} (ln \frac{16}{3} + \frac{2}{3} e^{x} - ln \frac{10}{3} + \frac{2}{3} e^{x})) + C

作图

流行的例子

limit as x approaches 0 of-x^2+3x+a

x \to 0 lim (- x^{2} + 3 x + a)

(dh)/(dt)=0.3(5-h)

\frac{d h}{d t} = 0.3 (5 - h)

integral of 36t(9t^2-7)^3

\int 36 t (9 t^{2} - 7)^{3} d t

laplacetransform 4sin(2t)

l a pl a ce t r an s f or m 4 sin (2 t)

integral from-1 to 1 of e^{-x}cos(npix)

\int_{- 1}^{1} e^{- x} cos (nπ x) d x