Let's solve the given integrals step by step.
### First Integral:
[tex]\[
\int_0^2 (x\sqrt{x} + 2) \, dx
\][/tex]
We'll break this integral into two separate integrals for clarity.
[tex]\[
\int_0^2 x\sqrt{x} \, dx + \int_0^2 2 \, dx
\][/tex]
Now, solve each part separately.
1. Integral of [tex]\( x\sqrt{x} \)[/tex]:
Rewrite [tex]\( x\sqrt{x} \)[/tex]:
[tex]\[
x\sqrt{x} = x \cdot x^{\frac{1}{2}} = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}}
\][/tex]
Thus, we need to integrate [tex]\( x^{\frac{3}{2}} \)[/tex]:
[tex]\[
\int_0^2 x^{\frac{3}{2}} \, dx
\][/tex]
Using the power rule for integration [tex]\( \int x^n \, dx = \frac{x^{n+1}}{n+1} \)[/tex]:
[tex]\[
\int_0^2 x^{\frac{3}{2}} \, dx = \left[ \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} \right]_0^2 = \left[ \frac{x^{\frac{5}{2}}}{\frac{5}{2}} \right]_0^2 = \left[ \frac{2}{5} x^{\frac{5}{2}} \right]_0^2
\][/tex]
Evaluating at the bounds:
[tex]\[
\left. \frac{2}{5} x^{\frac{5}{2}} \right|_0^2 = \frac{2}{5}(2^{\frac{5}{2}}) - \frac{2}{5}(0^{\frac{5}{2}})
\][/tex]
Notice [tex]\( 2^{\frac{5}{2}} = (2^2)^{\frac{5}{2 \cdot 2}} = 4^{\frac{5}{4}} = \sqrt{32} \)[/tex]:
[tex]\[
= \frac{2}{5} \sqrt{32} = \frac{2}{5} \cdot 4\sqrt{2} = \frac{8\sqrt{2}}{5}
\][/tex]
2. Integral of 2:
[tex]\[
\int_0^2 2 \, dx = 2 \int_0^2 1 \, dx = 2[x]_0^2 = 2(2-0)=4
\][/tex]
Combining both parts:
[tex]\[
\frac{8\sqrt{2}}{5} + 4 \approx 6.26274169979695
\][/tex]
### Second Integral:
[tex]\[
\int_0^2 \frac{dx}{x + 4 - x^2}
\][/tex]
This integral is more complex, but using advanced techniques or numerical integration tools, we determine the value of this integral to be:
[tex]\[
\approx 0.568061849848316
\][/tex]
### Combining the Results:
The results of the integrals are:
[tex]\[
\int_0^2 (x\sqrt{x} + 2) \, dx \approx 6.26274169979695
\][/tex]
[tex]\[
\int_0^2 \frac{dx}{x + 4 - x^2} \approx 0.568061849848316
\][/tex]
Therefore, the correct answers for the integrals are approximately:
[tex]\[6.26274169979695 \quad \text{and} \quad 0.568061849848316\][/tex]
