INTEGRATION

INTEGRATION

1. in mathematics, integration is a technique of finding a function g(x) the derivative of which, Dg(x), is equal to a given function f(x).This is indicated by the inseparable sign "∫," as in ∫f (x), usually called the integral of an infinite function.

   • DEFINITION

2. ∫ 1 dx = x + C

   • FORMULA
   • ∫ a dx = ax+ C
     • ∫ xn dx = ((xn+1)/(n+1))+C ; n≠1
     • ∫ sin x dx = – cos x + C
       • ∫ cos x dx = sin x + C

3. EXAMPLE

   • 1. What is the value of ∫ 8 x3 dx. Solution: ∫ 8 x3 dx = 8 ∫ x3 dx = 8 x4 / 4 + C = 2 x4 + C
     • 2.  Find the value of ∫ Cos x + x dx. Solution: ∫ Cos x + x dx = ∫ Cos x dx + ∫ x dx = sin x + x2/2 + C

4. APPLICATIONS

   • 1. Area between curves 2. Distance, Velocity, Acceleration
     • 3. Kinetic energy; improper integrals 4. Probability
       • 5. Average value of a function 6. Center of Mass