Integrals and Applications of Integrals
Integrals and Applications of Integrals tests indefinite and definite integration, standard forms, substitution, integration by parts, partial fractions and area under curves.
Core rules
- Indefinite integrals require +C; definite integrals are numbers and use limits.
- Standard forms include ∫x^n dx=x^(n+1)/(n+1) for n≠-1, ∫1/x dx=ln|x|, ∫e^x dx=e^x, ∫cos x dx=sin x.
- Substitution changes dx and limits together in a definite integral.
- Integration by parts: ∫u dv=uv-∫v du; choose u by the usual order of algebraic, trigonometric, exponential and logarithmic functions.
- Area between a curve and the x-axis is the integral of |y|; signed integrals count area below the axis as negative.
Worked example
Evaluate ∫ from 0 to 1 of x e^(x^2) dx. Let u=x^2, du=2x dx. When x=0, u=0; when x=1, u=1. The integral becomes (1/2)∫ from 0 to 1 e^u du=(1/2)(e-1). Changing limits avoids substituting back.
Traps
- Do not forget +C in indefinite integrals.
- Do not change the integrand to u but leave limits in x.
- Splitting an integral across a point where the function changes sign needs absolute value for geometric area.
- By parts requires choosing dv that is easy to integrate; swapping u and dv can make it worse.
- Check for improper integrals when the denominator vanishes inside the interval.