Question

Find the indefinite integral. (Use C for the constant of integration.) e2x 25 e4x dx.

Answer 1

Answer:

The solution is  $\frac{1}{10} * tan^{-1}[\frac{e^{2x}}{5} ] + C$

Step-by-step explanation:

From the question

    The function given is  $f(x) = \frac{e^{2x}}{ 25 + e^{4x}} dx$

The  indefinite integral is  mathematically represented as

          $\int\limits {\frac{e^{2x}}{ 25 + e^{4x}}} \, dx$

Now  let  $e^{2x} = u$

=>   $\frac{du}{dx} 2e^{2x}$

=>   $2 e^{2x}dx = du$

So

$\int\limits {\frac{e^{2x}}{ 25 + e^{4x}}} \, dx = \int\limits {\frac{1}{ 2(25 + u^2)} } \, du$

$= \frac{1}{2} \int\limits {\frac{1}{ 25 + u^2)} } \, du$

$= \frac{1}{2} \int\limits {\frac{1}{ 5^2 + u^2)} } \, du$

$= \frac{1}{2} \frac{tan^{-1} [\frac{u}{5} ]}{5} + C$

Now substituting for  u

$\frac{1}{10} * tan^{-1}[\frac{e^{2x}}{5} ] + C$