Question

$\int \frac{e^{\cot ^{-1} x}}{1+x^{2}}\left(x^{2}-x+1\right) d x=\ldots \ldots \ldots \ldots+c$

• A. $x \cdot e^{\cot ^{-1} x}$
• B. $e^{\cot ^{-1} x}$
• C. $\frac{e^{\cot ^{-1} x}}{1+x^{2}}$
• D. $-e^{\cot ^{-1} x}$

Answer 1

Answer: (A)

$\int \frac{e^{\cot ^{-1} x}}{1+x^{2}}\left(x^{2}-x+1\right) d x$
Let $\cot ^{-1} x=t$
$\Rightarrow -\frac{1}{1+x^{2}} d x=d t$
$\int e^{t}\left(\cot t-\text{cosec}^{2} t\right) d t$
$e^{t} \cot t+c$
$x e^{\cot ^{-1} x}+c$