Question

remember that it is not true that if f and g are antiderivatives of f and g, respectively, then f·g is an antiderivative of f·g. however, we can expand the product and use fractional exponents to rewrite f(x)=√x(9 + 5x) as x^{1/2}+x^{3/2}.

Explanation:

Step1: Expand the product

Multiply \(\sqrt{x}\) with \(9\) and \(5x\) separately.
When we multiply \(\sqrt{x}=x^{\frac{1}{2}}\) by \(9\), we get \(9x^{\frac{1}{2}}\).
When we multiply \(x^{\frac{1}{2}}\) by \(5x = 5x^{1}\), using the rule \(a^{m}\cdot a^{n}=a^{m + n}\), we have \(5x^{\frac{1}{2}+1}=5x^{\frac{3}{2}}\).

Answer:

\(9x^{1/2}+5x^{3/2}\)