Question

Evaluate $\int \frac{7x+2}{\sqrt{2x^2+5x}}dx$
(where $C$ is constant of integration)

• A. $-\frac{7}{2}\sqrt{2x^2+5x}-\frac{27}{8\sqrt{2}}\ln |(x+\frac{5}{4})+\sqrt{x^2+\frac{5x}{2}}|+C$
• B. $-\frac{7}{2}\sqrt{2x^2+5x}+\frac{31}{2\sqrt{2}}\ln |(x+\frac{5}{4})+\sqrt{x^2+\frac{5x}{2}}|+C$
• C. $\frac{7}{2}\sqrt{2x^2+5x}-\frac{31}{4\sqrt{2}}\ln |(x+\frac{5}{4})+\sqrt{x^2+\frac{5x}{2}}|+C$
• D. $\frac{7}{2}\sqrt{2x^2+5x}-\frac{27}{4\sqrt{2}}\ln |(x+\frac{5}{4})+\sqrt{x^2+\frac{5x}{2}}|+C$

Answer 1

The correct option is D $\frac{7}{2}\sqrt{2x^2+5x}-\frac{27}{4\sqrt{2}}\ln |(x+\frac{5}{4})+\sqrt{x^2+\frac{5x}{2}}|+C$

Let $I=\int \frac{7x+2}{\sqrt{2x^2+5x}}dx$
$\Rightarrow I=\frac{7}{4}\int \frac{4x+5}{\sqrt{2x^2+5x}}dx+(2-\frac{35}{4})\int \frac{dx}{\sqrt{2x^2+5x}}$
$\Rightarrow I=\frac{7}{4}\cdot 2\sqrt{2x^2+5x}-\frac{27}{4\sqrt{2}}\int \frac{dx}{\sqrt{x^2+\frac{5}{2}x}}$

$\Rightarrow I=\frac{7}{2}\sqrt{2x^2+5x}-\frac{27}{4\sqrt{2}}\int \frac{dx}{\sqrt{{(x+\frac{5}{4})}^2-{\left(\frac{5}{4}\right)}^2}}$

$\therefore I=\frac{7}{2}\sqrt{2x^2+5x}-\frac{27}{4\sqrt{2}}\ln |(x+\frac{5}{4})+\sqrt{x^2+\frac{5x}{2}}|+C$