What is the integral of (e^{4x})/(sqrt(25-e^{2x))} ?

The integral of (e^{4x})/(sqrt(25-e^{2x))} is -1/3 (e^{2x}+50)sqrt(-e^{2x)+25}+C

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integral of (e^{4x})/(sqrt(25-e^{2x))}

\int \frac{e ^{4 x}}{25 - e ^{2 x}} d x

- \frac{1}{3} (e^{2 x} + 50) - e^{2 x} + 25 + C

Solution steps

\int \frac{e ^{4 x}}{25 - e ^{2 x}} d x

Apply u-substitution

: \int - \frac{- u + 25}{2 u} d u

= \int - \frac{- u + 25}{2 u} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int \frac{- u + 25}{u} d u

Expand

\frac{- u + 25}{u} : - u + \frac{25}{u}

= - \frac{1}{2} \cdot \int - u + \frac{25}{u} d u

Apply the Sum Rule:

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

= - \frac{1}{2} (- \int u d u + \int \frac{25}{u} d u)

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int \frac{25}{u} d u = 50 u

= - \frac{1}{2} (- \frac{2}{3} u^{\frac{3}{2}} + 50 u)

Substitute back

u = 25 - e^{2 x}

= - \frac{1}{2} (- \frac{2}{3} (25 - e^{2 x})^{\frac{3}{2}} + 50 25 - e^{2 x})

Simplify

- \frac{1}{2} (- \frac{2}{3} (25 - e^{2 x})^{\frac{3}{2}} + 50 25 - e^{2 x}) : - \frac{1}{3} (e^{2 x} + 50) - e^{2 x} + 25

= - \frac{1}{3} (e^{2 x} + 50) - e^{2 x} + 25

Add a constant to the solution

= - \frac{1}{3} (e^{2 x} + 50) - e^{2 x} + 25 + C

\frac{\partial}{\partial x} (ln (x^{2} + 2 y^{2}))

\frac{d y}{d x} = \frac{y}{( 1 + x ) ( 2 + x )}

\frac{d}{d x} (\frac{x ^{2}}{4 - x ^{2}})

\frac{d}{d x} ((\frac{x - 2}{x + 2})^{2})

\frac{\partial}{\partial x} (40 x)

What is the integral of (e^{4x})/(sqrt(25-e^{2x))} ?
The integral of (e^{4x})/(sqrt(25-e^{2x))} is -1/3 (e^{2x}+50)sqrt(-e^{2x)+25}+C