To find the function [tex]\(f(x)\)[/tex] given that [tex]\( f'(x) = \sqrt{x}(6 + 10x) \)[/tex] and the initial condition [tex]\( f(1) = 9 \)[/tex], follow these steps:
1. Identify the integrand:
The derivative of the function is given by [tex]\( f'(x) = \sqrt{x}(6 + 10x) \)[/tex].
2. Integrate [tex]\( f'(x) \)[/tex] to find [tex]\( f(x) \)[/tex]:
To determine [tex]\( f(x) \)[/tex], we integrate [tex]\( f'(x) \)[/tex] with respect to [tex]\( x \)[/tex].
[tex]\[
f(x) = \int \sqrt{x}(6 + 10x) \, dx
\][/tex]
3. Break down the integrand:
Separate the integrand into two parts to simplify the integration:
[tex]\[
\int \sqrt{x}(6 + 10x) \, dx = \int 6\sqrt{x} \, dx + \int 10x^{3/2} \, dx
\][/tex]
4. Integrate each term separately:
- For the first term: [tex]\( \int 6\sqrt{x} \, dx = 6 \int x^{1/2} \, dx \)[/tex]
[tex]\[
6 \int x^{1/2} \, dx = 6 \cdot \frac{x^{3/2}}{3/2} = 6 \cdot \frac{2}{3} x^{3/2} = 4x^{3/2}
\][/tex]
- For the second term: [tex]\( \int 10x^{3/2} \, dx \)[/tex]
[tex]\[
10 \int x^{3/2} \, dx = 10 \cdot \frac{x^{5/2}}{5/2} = 10 \cdot \frac{2}{5} x^{5/2} = 4x^{5/2}
\][/tex]
5. Combine the integrated parts:
[tex]\[
f(x) = 4x^{3/2} + 4x^{5/2} + C
\][/tex]
Here, [tex]\( C \)[/tex] is the constant of integration.
6. Use the initial condition to determine [tex]\( C \)[/tex]:
Given [tex]\( f(1) = 9 \)[/tex]:
[tex]\[
9 = 4(1)^{3/2} + 4(1)^{5/2} + C
\][/tex]
Simplify the expression:
[tex]\[
9 = 4 \cdot 1 + 4 \cdot 1 + C = 8 + C \implies C = 9 - 8 = 1
\][/tex]
7. Write the final function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 4x^{3/2} + 4x^{5/2} + 1
\][/tex]
Thus, the function [tex]\( f(x) \)[/tex] is:
[tex]\[
\boxed{f(x) = 4x^{3/2} + 4x^{5/2} + 1}
\][/tex]
