Problem 8 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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

Step 3: 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. \((x^{2}-1)/x^{3 / 2}\)