Question

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int \frac{x^{2}-1}{x^{3 / 2}} d x=\frac{2\left(x^{2}+3\right)}{3 \sqrt{x}}+C $$

Short answer

Indeed, the derivative of the right side, when simplified, is equated to the integrand of the left side. Hence, the statement has been verified.

Step-by-step solution

Differentiate the right-hand side

Begin by differentiating the right-hand side of the equation, which is \(\frac{2(x^{2}+3)}{3\sqrt{x}}+C\). Use the quotient and chain rules for differentiation.

Quotient Rule

The quotient rule states that, for two functions u and v, where v ≠ 0, their derivative is: \((u/v)'=\left(vu'-uv'\right)/v^{2}\). Set \(u=2(x^{2}+3)\) and \(v=3x^{1/2}\), then find \(u'\) and \(v'\). \(u'=4x\) and \(v'=3/2x^{-1/2}\). Substituting into the quotient rule

Simplify the Expression

Upon substituting \(u\), \(u'\), \(v\) and \(v'\) into the quotient rule and simplifying, the result should be the integrand function on the left-hand side of the original equation, viz. \((x^{2}-1)/x^{3 / 2}\)