Integration and Applications
Core concepts
Candidates must integrate standard functions, use substitution (including trigonometric substitution), integration by parts, partial fractions before integrating, and evaluate definite integrals. Know the applications: area under a curve, area between curves, volumes of revolution, and simple differential equations by separation of variables.
Traps
- With definite integrals and substitution, change the limits or convert back to x; mixing the two is a common error.
- When finding area between curves, identify which curve is above and subtract accordingly; if the curves cross within the interval, split the integral.
- Integration by parts: choose u so its derivative is simpler; LIATE is a useful guide.
- For volumes of revolution, the formula uses y², and the limits must match the axis of rotation.
- Forgetting the constant of integration in indefinite integrals loses a mark.
Worked example
Evaluate ∫ x ln x dx. Use parts with u = ln x, dv = x dx: du = dx/x, v = x²/2. Then ∫ x ln x dx = (x²/2) ln x − ∫ (x²/2)(1/x) dx = (x²/2) ln x − ∫ x/2 dx = (x²/2) ln x − x²/4 + C. Rule of thumb: for a logarithm multiplied by a polynomial, let u be the logarithm.