Indefinite Integral Practice- Problems and Solutions
What You Actually Need to Know About Indefinite Integrals
You failed the last calculus test. Or you're about to. Either way, indefinite integrals are probably killing you. Here's the reality: integration is harder than differentiation. Period. The rules are messier, the intuition is harder to build, and there's no chain rule equivalent that works every time.
This guide cuts through the nonsense. You'll get real practice problems, actual solutions, and the straight truth about where students consistently screw up.
The Basic Formula You Should Already Know
An indefinite integral is the reverse of differentiation. If F'(x) = f(x), then:
∫f(x)dx = F(x) + C
The "+C" matters. Forgetting it is a guaranteed way to lose points. Every single time.
Essential Power Rule
For ∫xⁿ dx where n ≠ -1:
∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C
That's it. Apply it to any power except -1. For x⁻¹, you get ln|x| + C instead.
Common Indefinite Integral Formulas
- ∫k dx = kx + C (constant stays)
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
- ∫eˣ dx = eˣ + C
- ∫1/x dx = ln|x| + C
- ∫sin(x) dx = -cos(x) + C
- ∫cos(x) dx = sin(x) + C
- ∫sec²(x) dx = tan(x) + C
Memorize these. They're not optional.
Practice Problems and Solutions
Problem 1: Basic Power Rule
Solve: ∫x³ dx
Apply the power rule. Add 1 to the exponent: 3+1 = 4. Divide by the new exponent.
Answer: x⁴/4 + C
Problem 2: Constant Multiple
Solve: ∫5x² dx
Pull the constant out: 5 ∫x² dx
Apply power rule to x²: x³/3
Answer: 5(x³/3) + C = (5x³)/3 + C
Problem 3: Sum Rule
Solve: ∫(2x + 3x² - 4) dx
Integrate each term separately:
- ∫2x dx = x² + C
- ∫3x² dx = x³ + C
- ∫(-4) dx = -4x + C
Answer: x² + x³ - 4x + C
Problem 4: Exponential
Solve: ∫(eˣ + 2eˣ) dx
Simplify first: 3eˣ
The integral of eˣ is itself.
Answer: 3eˣ + C
Problem 5: Trig Function
Solve: ∫sin(x) dx
Standard formula: derivative of cos(x) is -sin(x), so derivative of -cos(x) is sin(x).
Answer: -cos(x) + C
Problem 6: Chain Rule in Reverse (u-Substitution)
Solve: ∫2x(x² + 1)³ dx
This screams substitution. Let u = x² + 1. Then du = 2x dx.
The integral becomes: ∫u³ du = u⁴/4 + C
Substitute back: (x² + 1)⁴/4 + C
Answer: (x² + 1)⁴/4 + C
Problem 7: Integration by Parts
Solve: ∫x·eˣ dx
Product of x and eˣ? Time for integration by parts: ∫u dv = uv - ∫v du
Pick u = x, dv = eˣ dx
Then du = dx, v = eˣ
Apply the formula:
= x·eˣ - ∫eˣ dx
= x·eˣ - eˣ + C
Answer: eˣ(x - 1) + C
Problem 8: Trig with Coefficient
Solve: ∫cos(3x) dx
You need substitution here. Let u = 3x, du = 3 dx, so dx = du/3.
∫cos(u) · (du/3) = (1/3)∫cos(u) du = (1/3)sin(u) + C
Answer: (1/3)sin(3x) + C
Integration Methods Compared
| Method | When to Use It | Key Sign |
|---|---|---|
| Power Rule | Simple powers of x | No composition, just x raised to a power |
| u-Substitution | Chain rule reversed | Composite function, nested expressions |
| Integration by Parts | Product of different function types | x times eˣ, x times sin(x), etc. |
| Trig Identities | Trig functions squared or multiplied | sin²(x), cos(x)sin(x) |
Where Students Actually Fail
Mistake 1: Forgetting the Constant
∫f(x) dx ≠ F(x)
It equals F(x) + C. Always. Every time. No exceptions.
Mistake 2: Wrong Substitution Choice
Students pick u = x² + 1 when it should be u = (x² + 1)³. Or vice versa. The inside of the most complex nested function is usually your u.
Mistake 3: Forgetting to Substitute Back
You solve in terms of u, then leave it in terms of u. Wrong. Every answer must be in terms of x.
Mistake 4: Integration by Parts Setup
LIATE exists for a reason. Log, Inverse trig, Algebraic, Trig, Exponential. Pick u from earlier in the list, dv from later. It doesn't always work perfectly, but it's your best starting point.
How to Actually Get Better
Read this guide once. Then close it. Here's what actually works:
- Do 20 problems minimum. Not 5. Not "I understand the examples." 20 different problems.
- Check your answers immediately. Integration has many possible forms. Use Wolfram Alpha or your textbook's answer key to verify.
- Identify which problems you can't start. Those are the ones where your substitution intuition is weak. Practice those specifically.
- Time yourself. If a basic power rule problem takes more than 30 seconds, you're too slow. Speed matters on exams.
The Brutal Truth
You can't memorize your way through integration. The formulas help, but you need pattern recognition. You need to see ∫x√(x²+1) dx and immediately think "u-substitution with u = x²+1."
That recognition only comes from practice. Not passive reading. Not watching videos. Active problem-solving, repeatedly, until the patterns stick.