To solve the integral [tex]\(\int\left(\frac{3 x^3 - x^2 + 7 x + 11}{\sqrt{x^3}}\right) dx\)[/tex], we can break it down into manageable parts by simplifying the integrand first.
The given integrand is:
[tex]\[
\frac{3 x^3 - x^2 + 7 x + 11}{\sqrt{x^3}}
\][/tex]
First, let's rewrite the integrand by breaking it down term by term:
[tex]\[
\frac{3 x^3 - x^2 + 7 x + 11}{\sqrt{x^3}} = \frac{3 x^3}{\sqrt{x^3}} - \frac{x^2}{\sqrt{x^3}} + \frac{7 x}{\sqrt{x^3}} + \frac{11}{\sqrt{x^3}}
\][/tex]
Next, simplify each term by factoring out the common power of [tex]\(x\)[/tex] from the numerator and denominator:
1. For the first term:
[tex]\[
\frac{3 x^3}{\sqrt{x^3}} = 3 x^{3/2}
\][/tex]
2. For the second term:
[tex]\[
\frac{x^2}{\sqrt{x^3}} = x^{2 - 3/2} = x^{1/2}
\][/tex]
3. For the third term:
[tex]\[
\frac{7 x}{\sqrt{x^3}} = 7 x^{1 - 3/2} = 7 x^{-1/2}
\][/tex]
4. For the fourth term:
[tex]\[
\frac{11}{\sqrt{x^3}} = 11 x^{-3/2}
\][/tex]
Now, rewrite the integral with these simplified terms:
[tex]\[
\int \left(3 x^{3/2} - x^{1/2} + 7 x^{-1/2} + 11 x^{-3/2} \right) dx
\][/tex]
We will now integrate each term separately:
1. Integrate [tex]\(3 x^{3/2}\)[/tex]:
[tex]\[
\int 3 x^{3/2} dx = 3 \cdot \frac{2}{5} x^{5/2} = \frac{6}{5} x^{5/2}
\][/tex]
2. Integrate [tex]\(-x^{1/2}\)[/tex]:
[tex]\[
\int -x^{1/2} dx = -\frac{2}{3} x^{3/2}
\][/tex]
3. Integrate [tex]\(7 x^{-1/2}\)[/tex]:
[tex]\[
\int 7 x^{-1/2} dx = 7 \cdot 2 x^{1/2} = 14 x^{1/2}
\][/tex]
4. Integrate [tex]\(11 x^{-3/2}\)[/tex]:
[tex]\[
\int 11 x^{-3/2} dx = 11 \cdot (-2) x^{-1/2} = -22 x^{-1/2}
\][/tex]
Combining all these results, we obtain the integrated expression:
[tex]\[
\frac{6}{5} x^{5/2} - \frac{2}{3} x^{3/2} + 14 x^{1/2} - 22 x^{-1/2}
\][/tex]
We initiate verification by re-expressing the integrated result in a simplified form:
[tex]\[
\frac{6x^4}{5\sqrt{x^3}} - \frac{2x^3}{3\sqrt{x^3}} + \frac{14x^2}{\sqrt{x^3}} - \frac{22x}{\sqrt{x^3}}
\][/tex]
Thus, the complete solution to the integral is:
[tex]\[
\int \left( \frac{3 x^3 - x^2 + 7 x + 11}{\sqrt{x^3}} \right) dx = \frac{6 x^4}{5 \sqrt{x^3}} - \frac{2 x^3}{3 \sqrt{x^3}} + \frac{14 x^2}{\sqrt{x^3}} - \frac{22 x}{\sqrt{x^3}} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
