Question

Evaluate the Integral \(\int {\frac{{{e^x}}}{{3 - {e^{2x}}}}dx} \).

Short answer

The value of the integral is \(\frac{1}{{2\sqrt 3 }}ln\left| {\frac{{{e^x} + \sqrt 3 }}{{{e^x} - \sqrt 3 }}} \right| + c\).

Substitute and then integrate it. After integration, again put \(t = {e^x}\)

Step-by-step solution

Substitute \({x^2} = t\).

Let \({e^x} = t\).

\({e^x}dx = dt\)

\(\int {\frac{{{e^x}}}{{3 - {e^{2x}}}}dx} = \int {\frac{{dt}}{{3 - {t^2}}}} \)

Evaluate the integral.

Using the formula, \(\int {\frac{{du}}{{{a^2} - {u^2}}} = \frac{1}{{2a}}ln\left| {\frac{{u + a}}{{u - a}}} \right| + c} \).

Hence, \(\int {\frac{{{e^x}}}{{3 - {e^{2x}}}}dx} = \frac{1}{{2\sqrt 3 }}ln\left| {\frac{{{e^x} + \sqrt 3 }}{{{e^x} - \sqrt 3 }}} \right| + c\).