Question

$\int \frac{2x+2}{\sqrt{x^2-4x-5}}dx=$

• A. $\sqrt{x^2-4x-5}+\log |x+\sqrt{x^2-4x-5}|+c$
• B. $\log \left|\sqrt{x^2-4x-5}\right|+\sqrt{x^2-4x-5}+c$
• C. $\sqrt{x^2-4x-5}+6\log |(x-2)+\sqrt{x^2-4x-5}|+c$
• D. $2\sqrt{x^2-4x-5}+6\log |(x-2)+\sqrt{x^2-4x-5}|$

Answer 1

The correct option is D $2\sqrt{x^2-4x-5}+6\log |(x-2)+\sqrt{x^2-4x-5}|$

$\int \frac{2x+2}{\sqrt{x^2-4x-5}}dx$

$=\int \frac{2(x+1)}{\sqrt{{(x+1)}^2-6(x+1)}}dx$

$=\int \frac{x+(x+2)}{\sqrt{{(x+2)}^2-9}}dx$

$=\int \frac{2(x-2)}{\sqrt{{(x+2)}^2-9}}$$+\frac{4}{\sqrt{{(x+2)}^2-9}}dx$

$=2\sqrt{x^2-4x-5}+6log|(x-2)+\sqrt{x^2-4x-5}|$