The integral provided in the image is:

$\int \frac{x^2}{\sqrt{x+5}} \, dx$

To solve this, we can use a substitution method.

Step-by-Step Solution:

1. Substitution: Let: $u = x + 5$ Therefore: $du = dx \quad \text{and} \quad x = u - 5$

2. Rewrite the Integral: Substitute $u = x + 5$ into the integral: $\int \frac{(u - 5)^2}{\sqrt{u}} \, du$

3. Expand the Numerator: Expand $(u - 5)^2$: $(u - 5)^2 = u^2 - 10u + 25$ So the integral becomes: $\int \frac{u^2 - 10u + 25}{\sqrt{u}} \, du$

4. Simplify the Expression: Divide each term in the numerator by $\sqrt{u}$: $\int \left( u^{\frac{3}{2}} - 10u^{\frac{1}{2}} + 25u^{-\frac{1}{2}} \right) \, du$

5. Integrate Each Term: Now integrate each term separately: $\int u^{\frac{3}{2}} \, du = \frac{2}{5} u^{\frac{5}{2}}$ $\int u^{\frac{1}{2}} \, du = \frac{2}{3} u^{\frac{3}{2}}$ $\int u^{-\frac{1}{2}} \, du = 2 u^{\frac{1}{2}}$

   Therefore, the integral becomes: $\frac{2}{5} u^{\frac{5}{2}} - \frac{20}{3} u^{\frac{3}{2}} + 50 u^{\frac{1}{2}} + C$

6. Substitute Back $u = x + 5$: Finally, substitute back $u = x + 5$: $\frac{2}{5} (x + 5)^{\frac{5}{2}} - \frac{20}{3} (x + 5)^{\frac{3}{2}} + 50 (x + 5)^{\frac{1}{2}} + C$

This is the final solution for the given integral.

Would you like further details or have any questions?

Related Questions:

1. How do you determine which substitution to use in integrals?
2. What are the common strategies for handling integrals with square roots?
3. Can we apply integration by parts to this problem? How would that work?
4. How would this integral change if the term inside the square root were more complex (e.g., $x^2 + 5$)?
5. What happens if we omit the constant of integration?

Tip:

When facing complex integrals, substitution is often a useful first step to simplify the integrand into more manageable terms.