What is the integral of (2e^{2x})/(e^{2x)+12e^x+35} ?

The integral of (2e^{2x})/(e^{2x)+12e^x+35} is ln|12e^x+e^{2x}+35|+6ln|e^x+7|-6ln|e^x+5|+C

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integral of (2e^{2x})/(e^{2x)+12e^x+35}

\int \frac{2 e ^{2 x}}{e ^{2 x} + 12 e ^{x} + 35} d x

ln 12 e^{x} + e^{2 x} + 35 + 6 ln ∣ e^{x} + 7 ∣ - 6 ln ∣ e^{x} + 5 ∣ + C

Solution steps

\int \frac{2 e ^{2 x}}{e ^{2 x} + 12 e ^{x} + 35} 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} + 12 e ^{x} + 35} d x

Apply u-substitution

= 2 \cdot \int \frac{u}{u ^{2} + 12 u + 35} d u

Complete the square

u^{2} + 12 u + 35 : (u + 6)^{2} - 1

= 2 \cdot \int \frac{u}{( u + 6 ) ^{2} - 1} d u

Apply u-substitution

= 2 \cdot \int \frac{v - 6}{v ^{2} - 1} d v

Expand

\frac{v - 6}{v ^{2} - 1} : \frac{v}{v ^{2} - 1} - \frac{6}{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{6}{v ^{2} - 1} d v)

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

\int \frac{6}{v ^{2} - 1} d v = - 6 (\frac{1}{2} ln ∣ v + 1 ∣ - \frac{1}{2} ln ∣ v - 1 ∣)

= 2 (\frac{1}{2} ln v^{2} - 1 - (- 6 (\frac{1}{2} ln ∣ v + 1 ∣ - \frac{1}{2} ln ∣ v - 1 ∣)))

Substitute back

= 2 (\frac{1}{2} ln (e^{x} + 6)^{2} - 1 - (- 6 (\frac{1}{2} ln ∣ e^{x} + 6 + 1 ∣ - \frac{1}{2} ln ∣ e^{x} + 6 - 1 ∣)))

Simplify

2 (\frac{1}{2} ln (e^{x} + 6)^{2} - 1 - (- 6 (\frac{1}{2} ln ∣ e^{x} + 6 + 1 ∣ - \frac{1}{2} ln ∣ e^{x} + 6 - 1 ∣))) : ln 12 e^{x} + e^{2 x} + 35 + 6 ln ∣ e^{x} + 7 ∣ - 6 ln ∣ e^{x} + 5 ∣

= ln 12 e^{x} + e^{2 x} + 35 + 6 ln ∣ e^{x} + 7 ∣ - 6 ln ∣ e^{x} + 5 ∣

Add a constant to the solution

= ln 12 e^{x} + e^{2 x} + 35 + 6 ln ∣ e^{x} + 7 ∣ - 6 ln ∣ e^{x} + 5 ∣ + C

d er i v a t i v e y = \frac{( 2 x ^{2} - 1 ) ^{3}}{( 5 x - 8 ) ^{2}}

\frac{d y}{d x} = 2 y

x \to 0 - lim (cos (x))

\frac{\partial}{\partial x} (2 y - x)

d er i v a t i v e cos (\frac{1 - e ^{8 x}}{1 + e ^{8 x}})

What is the integral of (2e^{2x})/(e^{2x)+12e^x+35} ?
The integral of (2e^{2x})/(e^{2x)+12e^x+35} is ln|12e^x+e^{2x}+35|+6ln|e^x+7|-6ln|e^x+5|+C