Common Mistakes
Traps in Definite & Indefinite Integrals
5 mistake patterns students fall for. Each one shows the wrong approach vs the correct approach.
Forgetting to change limits after substitution
FORMULAWhen using substitution in definite integrals, the limits must change to the new variable.
✗ WRONG: Substituting t = sinx but keeping limits as 0 to π/2 instead of converting to 0 to 1
✓ RIGHT: When t = sin(x): x=0 → t=0, x=π/2 → t=1. Use new limits with new variable.
Missing constant of integration
FORMULAEvery indefinite integral must have + C. Forgetting it in MCQs can lead to wrong option.
✗ WRONG: ∫2x dx = x² (without +C)
✓ RIGHT: ∫2x dx = x² + C. In MCQs, check if options differ by a constant.
Wrong application of King's property
FORMULAKing's property: ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx. The replacement is x → (a+b-x), not (b-a-x).
✗ WRONG: Replacing x with (b-x) when limits are a to b
✓ RIGHT: Replace x with (a+b-x). For 0 to π: replace with π-x. For 0 to 1: replace with 1-x.
Sign error in by-parts
SIGN ERROR∫u dv = uv - ∫v du. The minus sign before the second integral is often lost.
✗ WRONG: ∫x·eˣdx = x·eˣ + ∫eˣdx = x·eˣ + eˣ + C
✓ RIGHT: ∫x·eˣdx = x·eˣ - ∫eˣdx = x·eˣ - eˣ + C = eˣ(x-1) + C.
Leibniz rule: forgetting chain rule
FORMULAd/dx ∫₀^{g(x)} f(t)dt = f(g(x))·g'(x), not just f(g(x)).
✗ WRONG: d/dx ∫₀^{x²} sin(t)dt = sin(x²)
✓ RIGHT: d/dx ∫₀^{x²} sin(t)dt = sin(x²) · 2x. The g'(x) factor is essential.