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INTEGRATION

INTEGRATION. Most mathematical processes have inverses or opposites !. Exs. Add & subtract. Multiply & divide. x 2 & x. 2 x & log 2 x etc. The inverse of differentiation / finding the derivative. is integration / finding the integral. Notation.

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INTEGRATION

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  1. INTEGRATION Most mathematical processes have inverses or opposites ! Exs Add & subtract Multiply & divide x2 & x 2x & log2x etc. The inverse of differentiation / finding the derivative is integration / finding the integral

  2. Notation Derivatives F (x) = f(x) Integrals f(x) dx = F(x) + C see later(**) comes before formula dx comes after formula f(x) dx reads as “ the integral of f(x) dx”

  3. Process Differentiation mult by power reduce power by 1 Integration increase power by 1 divide by new power

  4. Consider If f(x) = 3x2 then f (x) = 6x So6x dx = 6x2 = 3x2 2 If g(x) = 1/10x10 then g (x) = x9 Sox9 dx = x10 = 1/10x10 10 Ex1 4x4 – 5x3 + 4x2 4 3 2 (4x3 – 5x2 + 4x) dx = = x4 – 5/3x3 + 2x2 + C

  5. Integrating Constants (Numbers) If y = 5x then dy/dx = 5 It follows that 5 dx = 5x In general k dx = kx + C (**) The Constant of Integration C Suppose f(x) = 5x2 – 4x + 7 and g(x) = 5x2 – 4x + 20 Then f (x) = 10x– 4 andg (x) = 10x - 4 NB: f (x) = g (x) but f(x)  g(x)

  6. This is because the constant terms are different ! Unless we have additional information we are not likely to know what the constant value is so we just call it C. Ex2 9x3 – 8x2 + 3x + C 3 2 = 3x3 – 4x2 + 3x + C ( 9x2 – 8x + 3) dx = Ex3 -6y-2 + C . -2  -6y-3 dy = = 3 + C …y2 = 3y -2 + C Ex4 6t1/2 dt = 6t3/2 + C 3/2 = 4t3/2 + C

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