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Additional Mathematics Theory           253

               For the function given ∂t/∂x = a, the constant terms becoming zero on
            differentiation. Integration is just the opposite of differentiation. While
            taking the differential of a function of one variable yields the gradient of
            a graph of y vs. x, integrating the function yields the area under the graph
            (from the curve to the y = 0 axis).
               Consider again the function y = a*x + b. The integral of y with respect
            to x is denoted by:

                                             2
                                     05
                                           *
                                                   +
                          *
                            + )
               Ú  y dx = ( a x b dx = . * a x +  bx c
                      Ú
            where c is a constant and the ∫ is like a drawn-out S, indicating that the
            summation is made over infinitessimally small increments of dx. Since
            integration is the opposite of differentiation:
                      )
               Ú ( dy dx dx =+
                            y c.
               The constant c arises because the gradient (dy/dx) contains no infor-
            mation about any fixed offset of y from the y = 0 axis (which disappears
            during differentiation).
               In order to determine the area under a graph of y vs. x, one needs to
            specify a start and end point for x. These are placed at the bottom and top
            of the ∫ sign. The integral becomes a definite integral. As with differenti-
            ation, most engineers have committed to memory the common indefinite
            integrals they are likely to need. Table A4.2 shows those commonly used.
               For a definite integral, the normal procedure is to first evaluate the
            indefinite integral, then subtract the value of the function at the start value
            from that at the end value to get the area under the graph. Hence, for our




                                        Table A4.2
            Function                                              ∫ ydx
                n
            y = x + a (n π-1)                              (1/n + 1)*x n+1  + a*x + c
                                                            x
            y = e x                                        e + c
            y = log(x) (log is natural logarithm base e)   x*(log(x) - 1) + c
            y = 1/x                                        log(x) + c
            y = sin(x) (x must be in radians = deg*p/180)  -cos(x) + c
            y = cos(x) (x must be in radians = deg*p/180)  sin(x) + c
            y = tan(x)                                     -log(cos(x)) + c
                                                            x
            y = a x                                        a /log(a) + c
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