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The equation is $\"\\int$ .

The integrand is $\"\\frac{x^2-1}{x^{\\frac{3}{2}}}\"$ .

Consider the right side part $\"f(x)=\\frac{2(x^2+3)}{3\\sqrt{x}}+C\"$ .

Apply derivative on each side with respect to $\"x\"$ .

$\"\\frac{d}{dx}f(x)=\\frac{d}{dx}\\left$

$\"f\'(x)=\\frac{d}{dx}\\left$

$\"=\\frac{2}{3}\\frac{d}{dx}\\left$

$\"=\\frac{2}{3}\\frac{d}{dx}\\left$

$\"=\\frac{2}{3}\\frac{d}{dx}\\left$

$\"=\\frac{2}{3}\\frac{d}{dx}\\left$

Apply power rule of derivatives $\"\\frac{d}{dx}x^{n}=nx^{n-1}\"$ .

$\"=\\frac{2}{3}\\left$

$\"=\\frac{2}{3}\\left$

$\"=\\frac{2}{3}\\left$

$\"=\\left$

$\"=\\left$

$\"=\\frac{x^{\\frac{1}{2}}x^{\\frac{3}{2}}-1}{x^{\\frac{3}{2}}}\"$

$\"=\\frac{x^{\\frac{1}{2}+\\frac{3}{2}}-1}{x^{\\frac{3}{2}}}\"$

$\"=\\frac{x^{\\frac{4}{2}}-1}{x^{\\frac{3}{2}}}\"$

$\"f\'(x)=\\frac{x^{2}-1}{x^{\\frac{3}{2}}}\"$ .

The derivative of right side is equal to the integrand.

$\"\\frac{d}{dx}\\left$ .

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$\"\\frac{d}{dx}\\left$ .