Question

निम्नलिखित प्रश्न को फलन का समाकलन कीजिए।

${\displaystyle \sqrt{x^2+3x}}$

Answer 1

माना ${\displaystyle I=\int \sqrt{x^2+3x}}$ dx

${\displaystyle =\int \sqrt{(x^2+3x+\frac{9}{4})-\frac{9}{4}}}$  dx

${\displaystyle =\int \sqrt{{(x+\frac{3}{2})}^2-{\left(\frac{3}{2}\right)}^2}}$ dx

${\displaystyle =\frac{(x+\frac{3}{2})}{2}\sqrt{{(x+\frac{3}{2})}^2-\frac{9}{4}}-\frac{9}{8} \log \left|(x+\frac{3}{2})+\sqrt{{(x+\frac{3}{2})}^2-\frac{9}{4}}\right|+C}$     ${\displaystyle ....[\because \int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\log |x+\sqrt{x^2-a^2}|+C]}$

${\displaystyle =\frac{2x+3}{4}\sqrt{x^2+3x}-\frac{9}{8} \log \left|x+\frac{3}{2}+\sqrt{x^2+3x}\right|+C}$