Solving a integral problem by using substitution method

Topic:- Integration

Integration is an important concept together with differentiation, forms one of the main operations in calculus help
Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral

is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.

Let's work out a simple example on this.
Question:-

solve ∫ ( x / √x²-9 )

Answer:-

We do it by substitution method.
 Let u = x2-9
    
    du
    ---- = 2x
    dx
 
   du
or ---- = xdx
   2
substituting these values ,the integral becomes

∫ (du/2) / √u

= 1/2 ∫ (u)-1/2 du

        (u)-1/2 + 1
= 1/2 ---------------
         -1/2 + 1
by equalizing the denominators
   
    -1+2
   ------- = 1/2
      2
So the integral becomes
    
        (u)1/2
= 1/2 ---------------   + c
         1/2

= (x2-9)1/2+C (as u =x2-9)

= √( x2-9 ) +c  is the Answer

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A problem on Integrating Factor

Topic:Integrating Factor

In simple term integrating factor in mathematics, is an function that is chosen to facilitate
the solving of a given equation involving differentials. It is generally used to solve ordinary differential equations, but is also used within multivitamin calculus,in this case often multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field).
In simple term integrating factor is denoted as I.F.

Question:
   
           dy
solve     ____ + = ex
           dx

Answer:

   Standard form: dy
                 ____ + p(x) * y = q(x)    Here p(x) = 1
                  dx                            q(x) = ex


 Now Inergrating factor = e√pdx = e√idx = ex

 Now the equation is:

                     y = i
                        ___
                        I.F  √I.F * Q(x) dx + c

Plugging in all the values:

                     y = 1
                        ___
                         ex √(ex * ex) dx + c

                     y = 1
                        ___
                         ex √(ex)2 dx + c

                     y = 1 * e2x
                        ___ ___ + c
                        (ex) 2

                     y = ex
                        ___ + c
                         2        

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