What is the integral of (2e^{2x})/(e^{2x)+10e^x+24} ?

The integral of (2e^{2x})/(e^{2x)+10e^x+24} is ln|10e^x+e^{2x}+24|+5ln|e^x+6|-5ln|e^x+4|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2e^{2x})/(e^{2x)+10e^x+24}

\int \frac{2 e ^{2 x}}{e ^{2 x} + 10 e ^{x} + 24} d x

ln 10 e^{x} + e^{2 x} + 24 + 5 ln ∣ e^{x} + 6 ∣ - 5 ln ∣ e^{x} + 4 ∣ + C

Solution steps

\int \frac{2 e ^{2 x}}{e ^{2 x} + 10 e ^{x} + 24} d x

Take the constant out:

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

= 2 \cdot \int \frac{e ^{2 x}}{e ^{2 x} + 10 e ^{x} + 24} d x

Apply u-substitution

: \int \frac{u}{u ^{2} + 10 u + 24} d u

= 2 \cdot \int \frac{u}{u ^{2} + 10 u + 24} d u

Complete the square

u^{2} + 10 u + 24 : (u + 5)^{2} - 1

= 2 \cdot \int \frac{u}{( u + 5 ) ^{2} - 1} d u

Apply u-substitution

: \int \frac{v - 5}{v ^{2} - 1} d v

= 2 \cdot \int \frac{v - 5}{v ^{2} - 1} d v

Expand

\frac{v - 5}{v ^{2} - 1} : \frac{v}{v ^{2} - 1} - \frac{5}{v ^{2} - 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{v}{v ^{2} - 1} d v - \int \frac{5}{v ^{2} - 1} d v)

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

\int \frac{5}{v ^{2} - 1} d v = - 5 (\frac{1}{2} ln ∣ v + 1 ∣ - \frac{1}{2} ln ∣ v - 1 ∣)

= 2 (\frac{1}{2} ln v^{2} - 1 - (- 5 (\frac{1}{2} ln ∣ v + 1 ∣ - \frac{1}{2} ln ∣ v - 1 ∣)))

Substitute back

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

Simplify

2 (\frac{1}{2} ln (e^{x} + 5)^{2} - 1 - (- 5 (\frac{1}{2} ln ∣ e^{x} + 5 + 1 ∣ - \frac{1}{2} ln ∣ e^{x} + 5 - 1 ∣))) : ln 10 e^{x} + e^{2 x} + 24 + 5 ln ∣ e^{x} + 6 ∣ - 5 ln ∣ e^{x} + 4 ∣

= ln 10 e^{x} + e^{2 x} + 24 + 5 ln ∣ e^{x} + 6 ∣ - 5 ln ∣ e^{x} + 4 ∣

Add a constant to the solution

= ln 10 e^{x} + e^{2 x} + 24 + 5 ln ∣ e^{x} + 6 ∣ - 5 ln ∣ e^{x} + 4 ∣ + C

- \frac{5}{x ^{2}}

\int 3 cos (\frac{π t}{2}) d t

\int 4 sin (x) cos (x) d x

x \to 0 lim (\frac{( sin ^{3} ( 2 x ) )}{( sin ^{3} ( 3 x ) )})

\int sin (x) (x) d x

What is the integral of (2e^{2x})/(e^{2x)+10e^x+24} ?
The integral of (2e^{2x})/(e^{2x)+10e^x+24} is ln|10e^x+e^{2x}+24|+5ln|e^x+6|-5ln|e^x+4|+C