Integration and Applications of Definite Integrals
Integration and Applications of Definite Integrals tests antiderivatives, standard forms, substitution, integration by parts, properties of definite integrals and area under or between curves. Accuracy in constants, limits and signs matters more than lengthy manipulation.
Core rules
- Standard results include ∫x^n dx = x^(n+1)/(n+1)+C for n ≠ -1, ∫(1/x)dx = ln|x|+C, ∫e^x dx=e^x+C, ∫cos x dx=sin x+C, ∫sin x dx=-cos x+C.
- For ∫f(g(x))g'(x)dx, substitute u=g(x), du=g'(x)dx. Change limits when the integral is definite.
- Integration by parts: ∫u dv = uv - ∫v du; choose u by the usual priority of inverse trig, log, algebraic, trig, exponential.
- For a definite integral, evaluate F(b)-F(a) after finding a consistent antiderivative.
- Area between y=f(x) and the x-axis from a to b is ∫|f(x)|dx; split where f changes sign.
- Area between two curves is ∫(upper - lower)dx over the intersection interval.
Worked example
Evaluate ∫_0^1 x e^(x^2) dx. Let u=x^2, so du=2x dx and x dx=du/2. When x=0, u=0; when x=1, u=1. The integral becomes (1/2)∫_0^1 e^u du = (1/2)(e-1). No arbitrary constant remains in a definite integral.
Mark-losing traps
- Forgetting the factor 1/2 from du=2x dx.
- Keeping old x-limits after substitution.
- Omitting absolute value when the curve lies below the x-axis.
- Adding +C to a definite integral.
- Using upper minus lower incorrectly after splitting an area.