Integrals and Application of Integrals
Integrals and Application of Integrals
This domain tests antiderivatives, definite integrals, and area. An indefinite integral represents a family of antiderivatives, so a constant C is required. A definite integral uses limits and gives a number; its value is signed area, so regions below the x-axis contribute negatively.
Core rules
- Standard forms: ∫x^n dx=x^{n+1}/(n+1)+C for n≠-1; ∫1/x dx=ln|x|+C; ∫e^x dx=e^x+C.
- Substitution reverses the chain rule; change limits when using a definite integral.
- Integration by parts: ∫u dv=uv-∫v du; choose u to simplify its derivative.
- Partial fractions handle rational integrands after checking degree.
- Area under y=f(x) from a to b is ∫_a^b |f(x)| dx; split where f changes sign.
- Area between curves is ∫(upper−lower) dx over the intersection interval.
Worked example
Evaluate ∫_0^1 x e^{x^2} dx. Let u=x^2, du=2x dx, so 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). Changing the limits avoids substituting back and is usually safer.
Traps
Omitting +C in indefinite integrals; forgetting to change limits after substitution; dropping a sign in integration by parts; treating signed area as geometric area; integrating across a vertical asymptote or a sign change without splitting.