Let’s solve the given indefinite integral using the substitution provided. The integral is:

$\int \frac{4x^3}{\sqrt{x^4 + 10}} \, dx$

and the substitution is $u = x^4 + 10$.

Step 1: Differentiate the substitution

If $u = x^4 + 10$, then: $\frac{du}{dx} = 4x^3 \quad \implies \quad du = 4x^3 \, dx.$

This means $4x^3 \, dx$ can be replaced by $du$ in the integral.

Step 2: Rewrite the integral

Using the substitution $u = x^4 + 10$, the integral becomes: $\int \frac{4x^3}{\sqrt{x^4 + 10}} \, dx = \int \frac{1}{\sqrt{u}} \, du.$

Step 3: Simplify the new integral

The integral $\int \frac{1}{\sqrt{u}} \, du$ can be written as: $\int u^{-\frac{1}{2}} \, du.$

Using the power rule for integration: $\int u^n \, du = \frac{u^{n+1}}{n+1} + C, \quad \text{where } n \neq -1,$ we get: $\int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}} + C.$

Step 4: Substitute back $u = x^4 + 10$

Returning to the original variable $x$, we substitute $u = x^4 + 10$ into the solution: $2u^{\frac{1}{2}} + C = 2\sqrt{x^4 + 10} + C.$

Final Answer:

$\int \frac{4x^3}{\sqrt{x^4 + 10}} \, dx = 2\sqrt{x^4 + 10} + C.$

Would you like a deeper explanation or have any questions?

Here are 5 related questions for practice:

1. Solve $\int \frac{x^3}{\sqrt{x^4 + 5}} \, dx$ using substitution.
2. Evaluate $\int \frac{1}{\sqrt{1 + x^2}} \, dx$.
3. How does substitution simplify integration?
4. Compute $\int x^3 \ln(x^4 + 10) \, dx$.
5. Differentiate $2\sqrt{x^4 + 10} + C$ to verify the solution.

Tip: Always check the bounds when solving definite integrals after substitution!