Show that -dx- 1-ex-e2x -log z-1-2 - z2-z-1 - where z-1-ex-

Let #160;I=∫ #160; e^ x dx / e^ x √(( 1+e^ x +e^ 2x ) #160;) #160; Put #160;t=e^ x ⇒ dt= e ^ x dx #160;I=∫ #160; dt / t√( 1+t+t^ 2 ...

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Standard XII

Mathematics

Integration by Substitution

Show that ∫...

Question

Show that $\int \frac{d x}{\sqrt{(1 + e^{x} + e^{2 x})}} = log {(z + \frac{1}{2}) + \sqrt{(z^{2} + z + 1)}}$ . where $z = \frac{1}{e^{x}}$ .

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Solution

Let $I = \int \frac{e^{x} d x}{e^{x} \sqrt{(1 + e^{x} + e^{2 x})}}$
Put $t = e^{x}$
$\Rightarrow d t = e^{x} d x$
$I = \int \frac{d t}{t \sqrt{1 + t + t^{2}}}$
Now put $t = \frac{1}{z} \Rightarrow d t = - \frac{1}{z^{2}} d z$
$I = - \int \frac{d z}{\sqrt{(z^{2} + z + 1)}}$

$= \int \frac{d z}{\sqrt{{(z + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}}}$
$I = log {(z + \frac{1}{2}) + \sqrt{(z^{2} + z + 1)}} + C$ where $z = \frac{1}{t} = \frac{1}{e^{x}}$

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