The problem provided illustrates an integration technique using algebraic substitution. Let's break down the approach shown for $\int (5x - 2)^{10} \, dx$:

Solution Process

1. Substitute: Let $u = 5x - 2$. Then, by differentiating with respect to $x$, we get: $\frac{du}{dx} = 5 \Rightarrow dx = \frac{du}{5}$

2. Rewrite the Integral: Substitute $u$ and $dx$ into the integral: $\int (5x - 2)^{10} \, dx = \int u^{10} \cdot \frac{1}{5} \, du$ This simplifies to: $\frac{1}{5} \int u^{10} \, du$

3. Integrate: Now integrate $u^{10}$ with respect to $u$: $\frac{1}{5} \cdot \frac{u^{11}}{11} + C = \frac{u^{11}}{55} + C$

4. Substitute Back: Replace $u$ with $5x - 2$: $\int (5x - 2)^{10} \, dx = \frac{(5x - 2)^{11}}{55} + C$

The result is: $\frac{(5x - 2)^{11}}{55} + C$

Additional Exercises

Two integrals are suggested for practice:

1. $\int \sqrt{3x + 4} \, dx$
2. $\int \frac{5}{(6 + x)^4} \, dx$

Would you like me to solve these exercises as well or provide further explanations on the substitution technique?

Expanding Knowledge

1. How can substitution be applied to integrals involving radicals like $\int \sqrt{3x + 4} \, dx$?
2. What are the common signs that suggest substitution as a method in integration?
3. How would you handle substitution when the integrand has multiple terms added or subtracted?
4. How does substitution differ from integration by parts?
5. Can you always use substitution when a function is raised to a power, or are there exceptions?

Tip: Always check that the derivative of your substitution expression matches part of the integrand exactly. This simplifies the process and avoids errors.