Integrals - BSEB Inter Study Guide
Integrals
This domain covers indefinite integrals, substitution, integration by parts, partial fractions and definite integrals using the fundamental theorem.
Core rules
- ∫x^n dx=x^(n+1)/(n+1)+C for n not equal to −1; ∫(1/x)dx=ln|x|+C.
- ∫e^x dx=e^x+C; ∫sin x dx=−cos x+C; ∫cos x dx=sin x+C.
- By parts: ∫u dv=uv−∫v du; choose u by LIATE (log, inverse trig, algebraic, trig, exponential).
- Definite integral: ∫ from a to b of f(x)dx=F(b)−F(a), where F′=f.
- Properties: ∫ from a to b = ∫ from a to c + ∫ from c to b; ∫ from 0 to a of f(x)dx=∫ from 0 to a of f(a−x)dx.
Worked example
Evaluate ∫x cos x dx. Take u=x, dv=cos x dx; then du=dx, v=sin x. ∫x cos x dx=x sin x−∫sin x dx=x sin x+cos x+C.
Traps
- Omitting +C in an indefinite integral.
- Forgetting to change limits after substitution in a definite integral.
- Choosing u and dv in the wrong order in by parts.
- Writing ∫f(x)g(x)dx=∫f∫g, which is false.
- Dropping the absolute value in ln|x|.