What is the integral of (e^{2x}-6e^x)/(e^{2x)+11} ?

The integral of (e^{2x}-6e^x)/(e^{2x)+11} is 1/2 ln|e^{2x}+11|-6/(sqrt(11))arctan(1/(sqrt(11))e^x)+C

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integral of (e^{2x}-6e^x)/(e^{2x)+11}

\int \frac{e ^{2 x} - 6 e ^{x}}{e ^{2 x} + 11} d x

\frac{1}{2} ln e^{2 x} + 11 - \frac{6}{11} arctan (\frac{1}{11} e^{x}) + C

Solution steps

\int \frac{e ^{2 x} - 6 e ^{x}}{e ^{2 x} + 11} d x

Expand

\frac{e ^{2 x} - 6 e ^{x}}{e ^{2 x} + 11} : \frac{e ^{2 x}}{e ^{2 x} + 11} - \frac{6 e ^{x}}{e ^{2 x} + 11}

Apply the Sum Rule:

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

= \int \frac{e ^{2 x}}{e ^{2 x} + 11} d x - \int \frac{6 e ^{x}}{e ^{2 x} + 11} d x

\int \frac{e ^{2 x}}{e ^{2 x} + 11} d x = \frac{1}{2} ln e^{2 x} + 11

\int \frac{6 e ^{x}}{e ^{2 x} + 11} d x = \frac{6}{11} arctan (\frac{1}{11} e^{x})

= \frac{1}{2} ln e^{2 x} + 11 - \frac{6}{11} arctan (\frac{1}{11} e^{x})

Add a constant to the solution

= \frac{1}{2} ln e^{2 x} + 11 - \frac{6}{11} arctan (\frac{1}{11} e^{x}) + C

f (x) = 20 x + 5

\int \frac{e ^{\frac{1}{x ^{4}}}}{x ^{5}} d x

x \to 2 lim (\frac{4 x - 3}{x + 2})

(\frac{x ^{4}}{4})^{^{'}}

\frac{360}{2 s ^{2} + 10 s + 8}

What is the integral of (e^{2x}-6e^x)/(e^{2x)+11} ?
The integral of (e^{2x}-6e^x)/(e^{2x)+11} is 1/2 ln|e^{2x}+11|-6/(sqrt(11))arctan(1/(sqrt(11))e^x)+C