Question

$\int\frac{dx}{x^2 +2x+2}$ equals
(A) $x\ tan^{-1}\ (x + 1) + C$
(B) $tan^{-1}\ (x + 1) + C$
(C) $(x + 1)\ tan^{-1}\ x + C$
(D) $tan^{-1}\x + C$

Answer 1

Solution:

To solve this try to form a complete square in the denominator of the function.

$\int\frac{dx}{x^2 +2x+2} \\ \\ =\int\frac{dx}{(x^2 +2x+1) + 1} \\ \\ = \int\frac{dx}{ {(x + 1)}^{2} + 1 }$
let
$x + 1 = t \\ \\ dx = dt \\ \\ substitute \\ \\ \int\frac{dt}{ {(t}^{2} + 1) } \\ \\ = {tan}^{ - 1} t + C \\ \\\int\frac{dx}{x^2 +2x+2} = {tan}^{ - 1} (x + 1) + C$
Option B is correct.

Hope it helps you.