Integration as the reverse of differentiation
Integration is the reverse of differentiation: if dF/dx = f(x) then ∫f(x)dx = F(x) + C, where C is the constant of integration. A definite integral has limits and gives a number, the area under the curve.
| Function | Integral |
|---|---|
| sin x | −cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
| eᵏˣ | eᵏˣ/k + C |
Methods of integration
| Method | When to use | Rule |
|---|---|---|
| Substitution | A function and its derivative appear together | Let u = g(x); replace dx by du/g′(x) |
| By parts | A product of two different functions | ∫u dv = uv − ∫v du |
| Partial fractions | A rational function | Split into simpler fractions first |
Worked exampleSubstitution
Find ∫ 2x(x² + 1)³ dx.
- Let u = x² + 1, so du = 2x dx.
- Integrate and substitute back.
Answer: The integral is (x² + 1)⁴ ÷ 4 + C.
Worked exampleBy parts
Find ∫ x eˣ dx.
- Take u = x and dv = eˣ dx, so v = eˣ.
- Integrate.
Answer: eˣ(x − 1) + C.
Worked examplePartial fractions
Find ∫ 1/((x − 1)(x + 1)) dx.
- Split.
- Integrate each.
Answer: The integral is ½ ln|(x − 1)/(x + 1)| + C.
Try itA definite integral
Choose the function a·xⁿ and the limits. The area under the curve from lo to hi.
Key terms
- Integration
- The reverse of differentiation.
- Definite integral
- An integral with limits, giving a number.
- Substitution
- Changing the variable to simplify an integral.
- Integration by parts
- Method for a product of functions.
Summary
- ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C.
- A definite integral gives the area under a curve.
- Methods: substitution, parts and partial fractions.
- Always add C to an indefinite integral.