Table of Contents
Definition
Integration is used to find areas, volumes, central points and often is used to find the area underneath the graph of a function.
Common Functions of Integration
Function | Integral | |
Constant | ∫ a dx | ax + c |
Variable | ∫ x dx | ![]() |
Square | ∫ x2 dx | ![]() |
Reciprocal | ∫ (1/x) dx | ln|x| + c |
Exponential | ∫ ex dx | ex + c |
∫ ax dx | ![]() | |
∫ ln(x) dx | x ln(x) − x + c | |
Trigonometry (x in radians) | ∫ cos(x) dx | sin(x) + c |
∫ sin(x) dx | -cos(x) + c | |
∫ sec2(x) dx | tan(x) + c |
Rules of Integration
Rules | Function | Integral |
Multiplication by constant | ∫ cf(x) dx | c∫f(x) dx |
Power Rule (n≠-1) | ∫ xn dx | ![]() |
Sum Rule | ∫ (f + g) dx | ∫f dx + ∫g dx |
Difference Rule | ∫ (f – g) dx | ∫f dx – ∫g dx |
Rules | Function | Application | Integral |
Multiplication by constant | ∫ 5 cos(x) dx | 5 ∫ cos(x) dx | 5 sin(x) + c |
Power Rule (n ≠ -1) | ∫ x2 dx | ![]() | ![]() |
Sum Rule | ∫ (cos(x) + x2) dx | ∫ cos(x) dx + ∫ x2 dx | ![]() |
Difference Rule | ∫ (x3 – sin(x)) dx | ∫ x3 dx – ∫ sin(x) dx | ![]() |
Examples






7.Find the area underneath the curve y = x2 + 3 from x = -1 to x = 1.
Answer:

8. Determine the volume of the solid of the revolution generated when the region is bounded by f(x) = -2×2 + 8 and the x -axis revolved about x-axis.
Answer:
Determine intersect point
-2x2 + 8 = 0
2x2 = 8
x2 = 4
x = 2 or x = -2
