Explanation

(d)
$\int_{}^{}\frac{x\ dx}{x^{2} + 4x + 5} = \int_{}^{}{\frac{x}{x^{2} + 4x + 4 + 1}dx}$
$= \int_{}^{}{\frac{x + 2 - 2}{(x + 2)^{2} + 1}dx}$
$= \frac{1}{2}\int_{}^{}{\frac{2(x + 2)dx}{(x + 2)^{2} + 1} - 2\int_{}^{}\frac{dx}{1 + (x + 2)^{2}}}$
$= \frac{1}{2}\log{\left\lbrack (x + 2)^{2} + 1 \right\rbrack - 2\tan^{- 1}{(x + 2) + c}}$
$= \frac{1}{2}\log{(x^{2} + 4x + 5) - 2\tan^{- 1}{(x + 2) + c}}$