Question

Evaluate: $\int \frac{dx}{\sqrt{x^2+3x-4}}$

• A. $ln\left|\frac{\sqrt{x+4}-\sqrt{x-1}}{\sqrt{x+4}+\sqrt{x-1}}\right|+c$
• B. $ln\left|\frac{\sqrt{x-4}+\sqrt{x-1}}{\sqrt{x+4}-\sqrt{x+1}}\right|+c$
• C. $ln\left|\frac{\sqrt{x+4}+\sqrt{x-1}}{\sqrt{x+4}-\sqrt{x-1}}\right|+c$
• D. None of these

Answer 1

The correct option is C $ln\left|\frac{\sqrt{x+4}+\sqrt{x-1}}{\sqrt{x+4}-\sqrt{x-1}}\right|+c$

Let $I=\int \frac{dx}{\sqrt{x^2+3x-4}}$
$I=\int \frac{dx}{\sqrt{x^2+\frac{3}{2}\left(2\right)x+\frac{9}{4}-4-\frac{9}{4}}}$
$I=\int \frac{dx}{\sqrt{{(x+\frac{3}{2})}^2-{\left(\frac{5}{2}\right)}^2}}$
$=\ln |(x+\frac{3}{2})+\sqrt{{(x+\frac{3}{2})}^2-{\left(\frac{5}{2}\right)}^2}|+C_1$
$=\ln |\frac{2x+3}{2}+\sqrt{x^2+3x-4}|+C_1$
$=\ln \left|\frac{(x-1)+(x+4)+2\sqrt{(x-1)(x+4)}}{2}\right|+C_1$
$=\ln \left|\frac{5{[\sqrt{x-1}+\sqrt{x+4}]}^2}{2[(x+4)-(x-1)]}\right|+C_1$
$=\ln \left(\frac{5}{2}\right)+\ln \left|\frac{{[\sqrt{x-1}+\sqrt{x+4}]}^2}{(\sqrt{x+4}-\sqrt{x-1})(\sqrt{x+4}+\sqrt{x-1})}\right|+C_1$
$=\ln \left|\frac{\sqrt{x+4}+\sqrt{x-1}}{\sqrt{x+4}-\sqrt{x-1}}\right|+C$