Question

$\int \left(\frac{\left(x^{2}+2\right)a^{\left(x +tan^{-1}x\right)}}{x^{2}+1}\right)dx =$

• A. $log a.a^{x+tan^{-1}x}+c$
• B. $\frac{\left(x+tan^{-1}x\right)}{log\,a}+c$
• C. $\frac{a^{x+tan^{-1}x}}{log\,a}+c$
• D. $log \,a(x + tan^{-1}x) + c$

Answer 1

Answer: (C)

$\frac{a^{x+tan^{-1}x}}{log\,a}+c$

Answer 2

Let $I=\int \frac{\left(x^{2}+2\right) a^{\left(x+\tan ^{-1} x\right)}}{x^{2}+1} dx$
Put $x+\tan ^{-1} x=t$
$\Rightarrow \left(1+\frac{1}{1+x^{2}}\right) dx=dt$
$\Rightarrow \frac{2+x^{2}}{1+x^{2}} dx=dt$
$\therefore I =\int a^{t} dt=\frac{a^{t}}{\log a}+c$
$=\frac{a^{x+\tan ^{-1} x}}{\log a}+c$