Question

Evaluate $\int \frac{x^2-3x+5}{\sqrt{x}}dx$

• A. $\frac{2}{5}{x}^{\frac{5}{2}}-2x^{\frac{3}{2}}+10\sqrt{x}+c$
• B. $\frac{5}{2}{x}^{\frac{5}{2}}-2x^{\frac{3}{2}}+10\sqrt{x}+c$
• C. $\frac{2}{5}{x}^{\frac{3}{2}}+2x^{\frac{3}{2}}+10\sqrt{x}+c$
• D. $\frac{2}{5}{x}^{\frac{3}{2}}-2x^{\frac{3}{2}}-10\sqrt{x}-c$

Answer 1

The correct option is A $\frac{2}{5}{x}^{\frac{5}{2}}-2x^{\frac{3}{2}}+10\sqrt{x}+c$

Let $I=\int \frac{x^2-3x+5}{\sqrt{x}}.dx$

$=\int \frac{x^2}{\sqrt{x}}dx-\int \frac{3x}{\sqrt{x}}dx+\int \frac{5}{\sqrt{x}}dx$

$=\int x^{\frac{3}{2}}dx-3\int x^{\frac{1}{2}}dx+5\int x^{\frac{-1}{2}}dx$

We know that, $\int x^n.dx=\frac{x^{n+1}}{n+1}$

$\Rightarrow I=\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}-3\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+5\frac{x^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+c$

$\therefore I=\frac{2}{5}{x}^{\frac{5}{2}}-2x^{\frac{3}{2}}+10\sqrt{x}+c$

$\int \frac{(x^2-3x+5)}{\sqrt{x}}dx=\frac{2}{5}{x}^{\frac{5}{2}}-2x^{\frac{3}{2}}+10\sqrt{x}+c$