Question

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

Explanation:

Step1: Integrate \(9x^{1/2}\)

Use the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). For \(n=\frac{1}{2}\), \(\int9x^{1/2}dx=9\times\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}=9\times\frac{x^{\frac{3}{2}}}{\frac{3}{2}} = 6x^{\frac{3}{2}}\)

Step2: Integrate \(5x^{3/2}\)

For \(n = \frac{3}{2}\), \(\int5x^{3/2}dx=5\times\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}=5\times\frac{x^{\frac{5}{2}}}{\frac{5}{2}}=2x^{\frac{5}{2}}\)

Answer:

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