What is the integral of (e^{5x})/((e^{5x)+11)^2} ?

The integral of (e^{5x})/((e^{5x)+11)^2} is -1/(5(e^{5x)+11)}+C

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integral of (e^{5x})/((e^{5x)+11)^2}

\int \frac{e ^{5 x}}{( e ^{5 x} + 11 ) ^{2}} d x

- \frac{1}{5 ( e ^{5 x} + 11 )} + C

Solution steps

\int \frac{e ^{5 x}}{( e ^{5 x} + 11 ) ^{2}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{e ^{5 x}}{( e ^{5 x} + 11 ) ^{2}} = (e^{5 x}) (e^{5 x} + 11)^{- 2}

= \int e^{5 x} (e^{5 x} + 11)^{- 2} d x

Expand

e^{5 x} (e^{5 x} + 11)^{- 2} : \frac{e ^{5 x}}{e ^{10 x} + 22 e ^{5 x} + 121}

= \int \frac{e ^{5 x}}{e ^{10 x} + 22 e ^{5 x} + 121} d x

Apply u-substitution

: \int \frac{1}{5 ( u ^{2} + 22 u + 121 )} d u

= \int \frac{1}{5 ( u ^{2} + 22 u + 121 )} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u ^{2} + 22 u + 121} d u

Take the partial fraction of

\frac{1}{u ^{2} + 22 u + 121} : \frac{1}{( u + 11 ) ^{2}}

= \frac{1}{5} \cdot \int \frac{1}{( u + 11 ) ^{2}} d u

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{v}

= \frac{1}{5} (- \frac{1}{v})

Substitute back

= \frac{1}{5} (- \frac{1}{e ^{5 x} + 11})

Simplify

\frac{1}{5} (- \frac{1}{e ^{5 x} + 11}) : - \frac{1}{5 ( e ^{5 x} + 11 )}

= - \frac{1}{5 ( e ^{5 x} + 11 )}

Add a constant to the solution

= - \frac{1}{5 ( e ^{5 x} + 11 )} + C

y^{^{'''}} - y^{^{''}} + 2 y^{^{'}} + 4 y = 0

\int_{- 1}^{1} e^{x} cos (x) d x

\frac{d}{d x} (x^{\frac{9}{5}} y^{7} z^{\frac{2}{7}})

y^{^{'}} = \frac{1}{10} y (500 - y)

x^{5}, x = 2

What is the integral of (e^{5x})/((e^{5x)+11)^2} ?
The integral of (e^{5x})/((e^{5x)+11)^2} is -1/(5(e^{5x)+11)}+C