Writing ∫₀² u³ du when the u-limits should be 1 to 5. Fix: the moment you write your new integral, immediately recompute both limits from u = g(x). Write them down before integrating. ✕
2. Forgetting du (treating dx as du)
Writing ∫(x²+1)⁵ dx → ∫u⁵ du when there was no 2x present. That’s wrong: you cannot substitute unless g′(x)dx is actually there. Fix: always write the line “du = … dx” explicitly and check it against the integrand. ✕
3. Choosing a poor u
Picking u = x⁵ for ∫2x(x²+1)⁵dx leaves you worse off. Fix: choose the inside of the composite, then verify its derivative is present. ✕
4. Leaving a stray x in a u-integral
e.g. ending up with ∫x√u du . Fix: rearrange your substitution to write x in terms of u (see Level 4 example above), or choose a different u. ✕
5. Forgetting to back-substitute
Leaving your final indefinite answer as u⁶/6 + c . The answer must be in terms of x. Fix: for indefinite integrals, the last line is always “replace u with …”. ✕
6. Losing the + c
Every indefinite integral needs + c . Free marks lost. (Definite integrals don’t need it: the c cancels.) ✓
Self-check in 5 seconds
Differentiate your answer. If you don’t get back the original integrand, something’s wrong. This is the fastest way to catch errors in the exam.