Integral Calculus
Integral calculus in Scholarship Calculus tests technique selection and justification: substitution, parts, partial fractions, trigonometric identities, improper integrals, and applications such as area and volume. The exam rewards recognising structure and checking whether an integral is proper before evaluating.
Core rules and concepts
- Substitution reverses the chain rule; change limits for definite integrals, or back-substitute.
- Integration by parts: ∫u dv = uv - ∫v du. Choose u to simplify under differentiation; LIATE is a guide, not a law.
- Partial fractions require degree(numerator) < degree(denominator); otherwise divide first.
- For improper integrals, replace the infinite limit or vertical asymptote with a parameter and take a limit.
- Area between curves uses ∫(upper-lower) dx; volume about the x-axis uses π∫y^2 dx.
Worked example
Evaluate ∫ from 1 to e of x ln x dx. Use parts with u=ln x, dv=x dx, so du=dx/x and v=x^2/2. Then ∫x ln x dx = (x^2/2)ln x - ∫(x^2/2)(1/x) dx = (x^2/2)ln x - x^2/4. Evaluate from 1 to e: at e, e^2/2 - e^2/4 = e^2/4; at 1, 0 - 1/4 = -1/4. Subtract: e^2/4 - (-1/4)= (e^2+1)/4.
Traps
Omitting dx in substitution; forgetting to change limits; losing the minus sign in parts; writing ∫1/x dx = ln x without absolute value; treating an improper integral as finite without checking convergence; using y^2 for volume about the y-axis incorrectly.