Question

step 1
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
9 9 x^{1/2}+5 5 x^{3/2}.
step 2
now, the most general antiderivative of f(x) = 9x^{1/2}+5x^{3/2} is
f(x)= + c.

Explanation:

Step1: Integrate each term separately

Use the power rule for integration \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)).
For the first - term \(\int9x^{1/2}dx\):
Let \(n=\frac{1}{2}\), then \(\int9x^{1/2}dx=9\int x^{1/2}dx\).
By the power rule, \(9\times\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}=9\times\frac{x^{3/2}}{\frac{3}{2}}=9\times\frac{2}{3}x^{3/2}=6x^{3/2}\).

Step2: Integrate the second term

For the second - term \(\int5x^{3/2}dx\):
Let \(n = \frac{3}{2}\), then \(\int5x^{3/2}dx=5\int x^{3/2}dx\).
By the power rule, \(5\times\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}=5\times\frac{x^{5/2}}{\frac{5}{2}}=5\times\frac{2}{5}x^{5/2}=2x^{5/2}\).

Answer:

\(6x^{3/2}+2x^{5/2}\)