Integration by substitution is a powerful method used to simplify integrals, especially when the integral involves a composition of functions. In substitution, you choose a part of the integrand to substitute with a new variable, simplifying the expression and making the integral easier to solve.

Here are a few example questions on integration by substitution:

1. Basic Substitution:

$\int (3x^2 + 2x) \, dx$ Hint: Choose $u = x^3 + x^2$, which simplifies the integral.

2. Trigonometric Substitution:

$\int \frac{1}{\sqrt{1 - x^2}} \, dx$ Hint: Use the substitution $x = \sin(\theta)$, so that $dx = \cos(\theta) \, d\theta$.

3. Exponential Substitution:

$\int e^{3x} \, dx$ Hint: Let $u = 3x$, then $du = 3dx$.

4. Rational Function Substitution:

$\int \frac{4x}{(x^2 + 1)^2} \, dx$ Hint: Use the substitution $u = x^2 + 1$, so that $du = 2x \, dx$.

5. Logarithmic Substitution:

$\int \frac{1}{x \ln(x)} \, dx$ Hint: Set $u = \ln(x)$, so that $du = \frac{1}{x} \, dx$.

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Would you like to go over the solutions to any of these or have more questions on substitution? Let me know if you'd like a detailed explanation or step-by-step breakdown for any of them!

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Related questions to explore:

1. How does the chain rule relate to integration by substitution?
2. What types of integrals are most commonly solved by substitution?
3. Can substitution help with definite integrals?
4. How do you handle the limits of integration when doing substitution?
5. What are some common mistakes to avoid during substitution?

Tip: When making a substitution, it’s important to also change the differential ($dx$ or $du$) accordingly after your substitution.