Question

$\int \frac{e^{2x}dx}{\sqrt[4]{4(e^x+1)}}=\frac{k}{21}{(e^x+1)}^{3/4}[3e^x-4].$ Find the value of $k$.

Answer 1

$I=\int \frac{e^x.e^{x}dx}{{(e^x+1)}^{1/4}}$
Put $e^x+1=t^4\therefore e^{x}dx=4t^{3}dt$
$\therefore I=\int \frac{(t^4-1).4t^3}{t}dt=4\int (t^5-t^2)dt$
$=4[\frac{t^2}{7}-\frac{t^3}{3}]=\frac{4t^3}{21}[3t^4-7]$
$=\frac{4}{21}{(e^x+1)}^{3/4}[3e^x-4].$