Sigma Tricks

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

Learn More

Integration by Substitution (Reverse Chain Rule)

Common Derivatives

Differentiable

Implicit Differentiation

Calculus Index