Question

Evaluate the integral
$\int \frac{2x+3}{\sqrt{x^2+4x+1}}dx$

• A. $2\sqrt{x^2+4x+1}-\log |x+2+\sqrt{x^2+4x+1}|+C$
• B. $\sqrt{x^2+4x-1}-\log |x+2+\sqrt{x^2+4x-1}|+C$
• C. $2\sqrt{x^2+4x+1}-\log |x-2+\sqrt{x^2-4x+1}|+C$
• D. $\sqrt{x^2+4x-1}-\log |x-2+\sqrt{x^2+4x-1}|+C$

Answer 1

The correct option is A $2\sqrt{x^2+4x+1}-\log |x+2+\sqrt{x^2+4x+1}|+C$

$I=\int \frac{2x+3}{\sqrt{x^2+4x+1}}dx=\int \frac{(2x+4)-1}{\sqrt{x^2+4x+1}}dx$
$=\int \frac{2x+4}{\sqrt{x^2+4x+1}}dx-\int \frac{1}{\sqrt{x^2+4x+1}}dx$
$=\int \frac{dt}{\sqrt{t}}-\int \frac{1}{\sqrt{{(x+2)}^2-{\left(\sqrt{3}\right)}^2}}dx,$ where $t=x^2+4x+1\Rightarrow dt=(2x+4)dx$
$=2\sqrt{t}-\log |(x+2)+\sqrt{x^2+4x+1}|+C$
$=2\sqrt{x^2+4x+1}-\log |x+2+\sqrt{x^2+4x+1}|+C$