Here's a problem set focusing on antidifferentiation (also known as integration), with answers included:

Problem Set on Antidifferentiation:

1. Find the indefinite integral: $\int 4x^3 \, dx$

2. Find the indefinite integral: $\int \left( \frac{1}{x^2} \right) dx$

3. Find the indefinite integral: $\int e^{2x} \, dx$

4. Find the indefinite integral: $\int \cos(x) \, dx$

5. Find the indefinite integral: $\int 3x^2 + 5x - 4 \, dx$

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Answers:

1. Solution for $\int 4x^3 \, dx$:

   $\int 4x^3 \, dx = \frac{4x^4}{4} + C = x^4 + C$

2. Solution for $\int \frac{1}{x^2} \, dx$:

   Rewrite $\frac{1}{x^2}$ as $x^{-2}$.

   $\int x^{-2} \, dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C$

3. Solution for $\int e^{2x} \, dx$:

   Use substitution, let $u = 2x$, then $du = 2dx$.

   $\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C$

4. Solution for $\int \cos(x) \, dx$:

   $\int \cos(x) \, dx = \sin(x) + C$

5. Solution for $\int (3x^2 + 5x - 4) \, dx$:

   $\int (3x^2 + 5x - 4) \, dx = \int 3x^2 \, dx + \int 5x \, dx - \int 4 \, dx$

   Compute each term: $\int 3x^2 \, dx = x^3$ $\int 5x \, dx = \frac{5x^2}{2}$ $\int 4 \, dx = 4x$

   Thus, the final answer is: $x^3 + \frac{5x^2}{2} - 4x + C$

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Further Exploration:

1. Can you explain how to apply substitution in antidifferentiation, as done in Problem 3?
2. How would you handle an integral involving a trigonometric function like $\sin(x)$?
3. What is the difference between indefinite and definite integrals, and how do you solve them?
4. How do you handle integrals of rational functions with higher powers in the denominator (e.g., $\int \frac{1}{x^n} \, dx$)?
5. What are some common techniques for solving integrals involving exponential functions?

Tip:

Always remember to add the constant of integration $C$ when working with indefinite integrals, as there can be multiple antiderivatives for a given function.