To evaluate the definite integral
[tex]\[
\int_4^7 \frac{8 x^2+3}{\sqrt{x}} \, dx,
\][/tex]
let's break it down step-by-step.
1. Rewrite the integrand: Start by expressing the integrand in a simpler form. Observe that
[tex]\[
\frac{8 x^2 + 3}{\sqrt{x}} = 8 x^{2} \cdot x^{-\frac{1}{2}} + 3 \cdot x^{-\frac{1}{2}} = 8 x^{\frac{3}{2}} + 3 x^{-\frac{1}{2}}.
\][/tex]
2. Integrate each term separately: Now isolate the terms to integrate them individually.
[tex]\[
\int (8 x^{\frac{3}{2}} + 3 x^{-\frac{1}{2}}) \, dx = \int 8 x^{\frac{3}{2}} \, dx + \int 3 x^{-\frac{1}{2}} \, dx.
\][/tex]
3. Integrate [tex]\(8 x^{\frac{3}{2}}\)[/tex]: Use the power rule for integration, which states
[tex]\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \, n \neq -1.
\][/tex]
So,
[tex]\[
\int 8 x^{\frac{3}{2}} \, dx = 8 \cdot \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = 8 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = 8 \cdot \frac{2}{5} x^{\frac{5}{2}} = \frac{16}{5} x^{\frac{5}{2}}.
\][/tex]
4. Integrate [tex]\(3 x^{-\frac{1}{2}}\)[/tex]: Similarly, applying the power rule,
[tex]\[
\int 3 x^{-\frac{1}{2}} \, dx = 3 \cdot \int x^{-\frac{1}{2}} \, dx = 3 \cdot \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = 3 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 3 \cdot 2 x^{\frac{1}{2}} = 6 x^{\frac{1}{2}}.
\][/tex]
5. Combine the results: The antiderivative of the integrand is thus
[tex]\[
\int (8 x^{\frac{3}{2}} + 3 x^{-\frac{1}{2}}) \, dx = \frac{16}{5} x^{\frac{5}{2}} + 6 x^{\frac{1}{2}} + C.
\][/tex]
6. Evaluate the definite integral: Now we need to evaluate this antiderivative from the lower limit 4 to the upper limit 7.
[tex]\[
\left[ \frac{16}{5} x^{\frac{5}{2}} + 6 x^{\frac{1}{2}} \right]_4^7 = \left( \frac{16}{5} (7)^{\frac{5}{2}} + 6 (7)^{\frac{1}{2}} \right) - \left( \frac{16}{5} (4)^{\frac{5}{2}} + 6 (4)^{\frac{1}{2}} \right).
\][/tex]
7. Compute each term separately:
- Evaluating at [tex]\( x = 7 \)[/tex]:
[tex]\[
\frac{16}{5} (7)^{\frac{5}{2}} + 6 (7)^{\frac{1}{2}} = \frac{16}{5} (343 \sqrt{7}) + 6 \sqrt{7}.
\][/tex]
- Simplify it:
[tex]\[
= \frac{5488 \sqrt{7}}{5} + 6 \sqrt{7} = \frac{5488 \sqrt{7}}{5} + \frac{30 \sqrt{7}}{5} = \frac{5488 \sqrt{7} + 30 \sqrt{7}}{5} = \frac{5518 \sqrt{7}}{5}.
\][/tex]
- Evaluating at [tex]\( x = 4 \)[/tex]:
[tex]\[
\frac{16}{5} (4)^{\frac{5}{2}} + 6 (4)^{\frac{1}{2}} = \frac{16}{5} \cdot 32 + 6 \cdot 2 = \frac{512}{5} + 12 = \frac{512 + 60}{5} = \frac{572}{5}.
\][/tex]
8. Subtract the evaluated results:
[tex]\[
\left( \frac{5518 \sqrt{7}}{5} \right) - \left( \frac{572}{5} \right) = \frac{5518 \sqrt{7} - 572}{5}.
\][/tex]
Thus, the result of the definite integral
[tex]\[
\int_4^7 \frac{8 x^2 + 3}{\sqrt{x}} \, dx
\][/tex]
is:
[tex]\[
\boxed{\frac{5518 \sqrt{7} - 572}{5}}.
\][/tex]
