Calculus
What this domain tests
Calculus carries a large mark share: differentiation by product, quotient, chain, implicit and parametric rules; integration by substitution, parts and partial fractions; definite integrals, areas and volumes of revolution; and first-order differential equations solved by separating variables.
Core rules
- d/dx(ln x) = 1/x, d/dx(e^{kx}) = ke^{kx}; the chain rule applies inside every composite.
- Integration by parts: ∫u dv = uv - ∫v du. Pick u to simplify when differentiated (ln x, x) and dv as what integrates easily.
- With a substitution in a definite integral, change the limits as well, or back-substitute before evaluating.
- Volume about the x-axis: V = π∫y² dx.
- Separable DE: collect y with dy and x with dx, integrate both sides, then use the initial condition to find the constant.
Worked example
Find ∫x ln x dx. Take u = ln x and dv = x dx, so du = (1/x)dx and v = x²/2. ∫x ln x dx = (x²/2)ln x - ∫(x²/2)(1/x)dx = (x²/2)ln x - (1/2)∫x dx = (x²/2)ln x - x²/4 + C. Check: differentiating gives x ln x + x/2 - x/2 = x ln x, as required.
Traps
- Dropping +C in indefinite integrals - a guaranteed lost mark.
- Losing the minus sign in ∫v du.
- Keeping x-limits after substituting u in a definite integral.
- Forgetting the dy/dx factor when differentiating implicitly.
- Differentiating trigonometric functions in degree mode; calculus needs radians.