Substitution is a great technique that helps us deal with complicated integrals. It makes them easier to understand and solve. The main idea is to replace a tricky part of the integral with a simpler letter, often making the entire problem easier.

How Substitution Works

1. Find a Substitution: Look for a part of the integral that you can replace to make it simpler. For example, in the integral $\int (2x)(x^2 + 1)^5 \, dx$, you can let $u = x^2 + 1$.

2. Calculate the Derivative: Next, find $du$. In our case, if $u = x^2 + 1$, then $du = 2x \, dx$.

3. Rewrite the Integral: Now, replace $u$ and $du$ in the integral. Our integral changes to: $\int u^5 \, du$.

4. Integrate: Then, we integrate with respect to $u$: $\frac{u^6}{6} + C$.

5. Back-Substitute: Finally, let’s switch back to the original variable: $\frac{(x^2 + 1)^6}{6} + C$.

Conclusion

By using substitution, we changed a complicated integral into a much simpler one. This shows how powerful this technique can be in math. It helps us solve tougher problems and understand math better as we move forward!