Below are 10 antiderivative practice problems in three levels: warm-up (basic rules), core (powers, roots and u-substitution) and challenge (integration by parts, partial fractions and trig identities). Each problem has a hint and a full step-by-step solution you can open after you try it, plus two bonus problems.
The fastest way to get good at antiderivatives is to practice the right problems in the right order. This set starts easy and builds up, so each level uses the skills from the one before. Try every problem on paper first. Open the hint only if you are stuck, and open the solution only after you have an answer.
How to Use This Page
- Work on paper. Writing the steps is what builds the skill.
- Check by differentiating. Before you open a solution, take the derivative of your answer. If you get the original function back, you are right.
- Score yourself. Use the guide at the bottom to see what to review next.
Level 1: Warm-Up (Basic Rules)
Problem 1
\[\int \left(4x^3 – 6x + 2\right)dx\]
Hint
Use the power rule on each term. Remember \(2 = 2x^0\).
Solution
\(\int 4x^3\,dx = x^4\), \(\int 6x\,dx = 3x^2\), \(\int 2\,dx = 2x\). Answer: \(x^4 – 3x^2 + 2x + C\).
Problem 2
\[\int \left(\frac{3}{x} + 2e^x\right)dx\]
Hint
\(\frac1x\) is the one power where the power rule does not work.
Solution
\(\int \frac3x\,dx = 3\ln|x|\) and \(\int 2e^x\,dx = 2e^x\). Answer: \(3\ln|x| + 2e^x + C\).
Problem 3
\[\int \left(5\cos x – \sec^2 x\right)dx\]
Hint
Which function has \(\sec^2 x\) as its derivative?
Solution
\(\int 5\cos x\,dx = 5\sin x\) and \(\int \sec^2 x\,dx = \tan x\). Answer: \(5\sin x – \tan x + C\).
Level 2: Core (Rewriting and Substitution)
Problem 4
\[\int \left(\sqrt{x} + \frac{1}{x^2}\right)dx\]
Hint
Rewrite as \(x^{1/2} + x^{-2}\) first.
Solution
\(\int x^{1/2}\,dx = \frac23x^{3/2}\) and \(\int x^{-2}\,dx = -x^{-1}\). Answer: \(\frac23x^{3/2} – \frac1x + C\).
Problem 5
\[\int (2x + 1)^5\,dx\]
Hint
Do not expand. Let \(u = 2x + 1\).
Solution
\(du = 2\,dx\), so \(dx = \frac12du\). Then \(\frac12\int u^5\,du = \frac{u^6}{12}\). Answer: \(\frac{(2x+1)^6}{12} + C\).
Problem 6
\[\int x\left(x^2 + 3\right)^4\,dx\]
Hint
The \(x\) outside is half the derivative of \(x^2 + 3\).
Solution
Let \(u = x^2 + 3\), \(du = 2x\,dx\). Then \(\frac12\int u^4\,du = \frac{u^5}{10}\). Answer: \(\frac{(x^2+3)^5}{10} + C\).
Problem 7
\[\int \sin x\,e^{\cos x}\,dx\]
Hint
What is the derivative of \(\cos x\)?
Solution
Let \(u = \cos x\), so \(du = -\sin x\,dx\). Then \(-\int e^u\,du = -e^u\). Answer: \(-e^{\cos x} + C\).
Level 3: Challenge (Advanced Methods)
Problem 8
\[\int x e^{-x}\,dx\]
Hint
Integration by parts with \(u = x\).
Solution
\(u = x\), \(dv = e^{-x}\,dx\), so \(du = dx\) and \(v = -e^{-x}\). Then \(-xe^{-x} + \int e^{-x}\,dx = -xe^{-x} – e^{-x}\). Answer: \(-e^{-x}(x + 1) + C\).
Problem 9
\[\int \frac{x + 3}{x^2 – x – 2}\,dx\]
Hint
Factor the bottom: \(x^2 – x – 2 = (x – 2)(x + 1)\). Then use partial fractions.
Solution
Write \(\frac{x+3}{(x-2)(x+1)} = \frac{A}{x-2} + \frac{B}{x+1}\). Setting \(x = 2\) gives \(A = \frac53\); setting \(x = -1\) gives \(B = -\frac23\). Answer: \(\frac53\ln|x – 2| – \frac23\ln|x + 1| + C\).
Problem 10
\[\int \cos^2(3x)\,dx\]
Hint
Use \(\cos^2\theta = \frac{1 + \cos 2\theta}{2}\) with \(\theta = 3x\).
Solution
\(\cos^2(3x) = \frac12 + \frac12\cos 6x\). Integrate: \(\frac{x}{2} + \frac{\sin 6x}{12}\). Answer: \(\frac{x}{2} + \frac{\sin 6x}{12} + C\).
Bonus Problems
Bonus A: find the exact function
Find \(f(x)\) if \(f'(x) = e^x + 2x\) and \(f(0) = 4\).
Solution
\(f(x) = e^x + x^2 + C\). Then \(f(0) = 1 + C = 4\), so \(C = 3\). Answer: \(f(x) = e^x + x^2 + 3\). See finding C.
Bonus B: a definite integral
\[\int_0^{\pi/2} \sin x\cos x\,dx\]
Solution
With \(u = \sin x\), the antiderivative is \(\frac{\sin^2 x}{2}\). Evaluate: \(\frac{1}{2} – 0 = \frac12\).
Score Yourself
| Score | What it means | Review next |
|---|---|---|
| 0–3 | The basic rules need more practice | Antiderivative rules and the power rule |
| 4–6 | Rules are solid; substitution is next | U-substitution patterns |
| 7–9 | Strong. Polish the advanced methods | Integration by parts |
| 10–12 | Excellent. You are ready for tests | Common mistakes to avoid under time pressure |
Want More Practice?
Make your own problems: change one number in any problem above, solve it, then check it in the antiderivative calculator. It shows every step, so you can see exactly where your work differs. For a mixed set that forces you to choose the method, try the quiz in how to find the antiderivative.
Frequently Asked Questions
How can I practice antiderivatives effectively?
Work from basic rules to advanced methods, write every step, and check each answer by differentiating. Mixing problem types once you know the methods helps you learn to choose the right one.
How do I know if my antiderivative is correct?
Differentiate it. If the derivative equals the function you started with, the answer is correct, even if it looks different from the answer key.
Why does my answer look different from the solution?
Two correct antiderivatives can differ by a constant or be written in different but equal forms. Differentiate both to compare them.
Which antiderivative problems appear most on tests?
Polynomial and root problems using the power rule, u-substitution problems, and integration by parts with \(x e^x\) or \(x\sin x\) appear most often in first-year calculus.
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