10 Common Integration Mistakes (And How to Avoid Them)
These errors cost students points on nearly every calculus exam. Learn to recognize and avoid them before your next test.
Mistake #1: Forgetting the + C
❌ Wrong
✅ Correct
Why it matters: The "+ C" represents all possible antiderivatives. Without it, your answer is incomplete for indefinite integrals.
Exception: Definite integrals don't need + C (the constants cancel out).
Mistake #2: Wrong Power Rule
❌ Wrong
✅ Correct
Remember: Add 1 to the exponent AND divide by the new exponent. Many students forget the division.
Mistake #3: Integrating 1/x Incorrectly
❌ Wrong
✅ Correct
Key insight: The power rule doesn't work for n = -1. This is a special case that gives natural log.
Mistake #4: Missing Absolute Value in ln
❌ Wrong
✅ Correct
Why: ln(x) is only defined for x > 0, but 1/x exists for all x ≠ 0. The absolute value extends the domain.
Mistake #5: Incorrect Sign for Trig Functions
❌ Wrong
✅ Correct
Memory tip: Verify by differentiating: d/dx[-cos(x)] = sin(x) ✓
Mistake #6: Forgetting Chain Rule Adjustment
❌ Wrong
✅ Correct
Rule: When the argument has a coefficient, divide by that coefficient after integrating.
Mistake #7: Not Simplifying Before Integrating
❌ Hard Way
✅ Easy Way
Always: Simplify algebraically before integrating when possible.
Mistake #8: Wrong Substitution Back
❌ Wrong
After substituting u = x², getting ∫u du = u²/2, then writing:
✅ Correct
Mistake #9: Incorrect Limits in Definite Integrals
❌ Wrong
Using u-sub on ∫₀¹ ... and keeping limits as 0 and 1
✅ Correct
Either change limits to u-values OR substitute back before evaluating
Mistake #10: Integration by Parts — Wrong Choice of u
❌ Bad Choice
For ∫x eˣ dx, choosing u = eˣ (makes it harder!)
✅ Good Choice
Use LIATE: u = x (Algebraic beats Exponential)
The Verification Trick
Always Check Your Answer!
Differentiate your result. You should get back the original integrand.
If ∫f(x)dx = F(x) + C, then F'(x) = f(x)