To find the general indefinite integral, we start by simplifying the integrand.

Given: $\int (x^5)^{4/2} \, dx$

Step 1: Simplify the exponent

$(x^5)^{4/2} = (x^5)^2 = x^{10}$

The integral simplifies to: $\int x^{10} \, dx$

Step 2: Apply the power rule of integration

The power rule states: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1$

Here, $n = 10$. Substituting: $\int x^{10} \, dx = \frac{x^{10+1}}{10+1} + C = \frac{x^{11}}{11} + C$

Final Answer:

$\frac{x^{11}}{11} + C$

Would you like further details or examples?

---

Here are 5 related questions to expand understanding:

1. How would you integrate $(x^3)^6$?
2. What happens if the integrand has fractional exponents, e.g., $(x^2)^{3/2}$?
3. How do you handle definite integrals of polynomials?
4. What is the general method to integrate $x^n$ when $n \neq -1$?
5. Can you find the derivative of $\frac{x^{11}}{11} + C$ to verify the result?

Tip: Simplify the integrand as much as possible before applying integration rules—it often reduces complexity significantly!