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  • Integrate the functions in Exercises 1 to 24.13

Integrate the functions in Exercises 1 to 24.

    Q13.    \frac{e^x}{(1 + e^x)(2 + e^x)}

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best_answer

we have to integrate the following function

\frac{e^x}{(1 + e^x)(2 + e^x)}

Let  1+e^x = t \implies e^xdx = dt

using this we can write the integral as 

\therefore \int\frac{e^x}{(1 + e^x)(2 + e^x)}dx = \int\frac{1}{t(1+t)}dt = \int\frac{(1+t)-t}{t(1+t)}dt

\\ = \int\left ( \frac{1}{t}-\frac{1}{t+1} \right )dt

\\ = \int\frac{1}{t}dt - \int\frac{1}{t+1}dt

\\ = \log t - \log (1+t) + C \\ = \log (1+e^x) - \log (2+e^x) + C \\ = \log\left ( \frac{e^x + 1}{e^x + 2} \right ) + C

Posted by

HARSH KANKARIA

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