Integrals and Application of Integrals
Integrals and Application of Integrals tests standard antiderivatives, substitution, integration by parts, partial fractions, definite integral properties, and area under curves. Definite integrals require limits and correct signs.
Core rules
Know standard forms such as ∫xⁿdx=xⁿ⁺¹/(n+1)+C for n≠−1, ∫1/x dx=log|x|+C, and ∫eˣdx=eˣ+C. For substitution, change dx and the limits together in a definite integral. Integration by parts uses ∫u dv=uv−∫v du, often choosing u by the LIATE order. For rational functions, decompose into partial fractions before integrating. Useful definite property: ∫₀ᵃ f(x)dx=∫₀ᵃ f(a−x)dx. Area above the x-axis is ∫ₐᵇ f(x)dx; area between curves is ∫ₐᵇ (upper−lower)dx.
Worked example
Evaluate I=∫₀^{π/2} sin x/(sin x+cos x) dx. Apply x→π/2−x: I=∫₀^{π/2} cos x/(cos x+sin x) dx. Adding the two forms gives 2I=∫₀^{π/2}1 dx=π/2, so I=π/4. The symmetry converts an awkward integrand into a constant.
Traps
Do not omit +C for indefinite integrals. When substituting u=g(x), do not leave old limits in a definite integral. Area below the x-axis gives a negative integral, so use absolute value for geometric area. Check whether a curve crosses the axis before writing a single integral. Do not confuse log|x| with log x when x may be negative.