Mathematics | Indefinite Integrals



Antiderivative –

  • Defination :A function ∅(x) is called the antiderivative (or an integral) of a function f(x) of ∅(x)’ = f(x).
  • Example : x4/4 is an antiderivative of x3 because (x4/4)’ = x3.
    In general, if ∅(x) is antiderivative of a function f(x) and C is a constant.Then,
    {∅(x)+C}' = ∅(x) = f(x).

Indefinite Integrals –

  • Defination :Let f(x) be a function. Then the family of all ist antiderivatives is called the indefinite integral of a function f(x) and it is denoted by ∫f(x)dx.
    The symbol ∫f(x)dx is read as the indefinite integral of f(x) with respect to x.
    Thus ∫f(x)dx= ∅(x) + C.
    Thus, the process of finding the indefinite integral of a function is called integration of the function.

Fundamental Integration Formulas –

  1. ∫xndx = (xn+1/(n+1))+C
  2. ∫(1/x)dx = (loge|x|)+C
  3. ∫exdx = (ex)+C
  4. ∫axdx = ((ex)/(logea))+C
  5. ∫sin(x)dx = -cos(x)+C
  6. ∫cos(x)dx = sin(x)+C
  7. ∫sec2(x)dx = tan(x)+C
  8. ∫cosec2(x)dx = -cot(x)+C
  9. ∫sec(x)tan(x)dx = sec(x)+C
  10. ∫cosec(x)cot(x)dx = -cosec(x)+C
  11. ∫cot(x)dx = log|sin(x)|+C
  12. ∫tan(x)dx = log|sec(x)|+C
  13. ∫sec(x)dx = log|sec(x)+tan(x)|+C
  14. ∫cosec(x)dx = log|cosec(x)-cot(x)|+C

Examples –

  • Example 1.Evaluate ∫x4dx.
  • Solution –
    Using the formula, ∫xndx = (xn+1/(n+1))+C
     ∫x4dx = (x4+1/(4+1))+C 
           = (x5/(5))+C  
    
  • Example 2.Evaluate ∫2/(1+cos2x)dx.
  • Solution –
    As we know that 1+cos2x = 2cos2x
    ∫2/(1+cos2x)dx = ∫(2/(2cos2x))dx
                   = ∫sec2x
                   = tan(x)+C  
    
  • Example 3.Evaluate ∫((x3-x2+x-1)/(x-1))dx.
  • Solution –
    ∫((x3-x2+x-1)/(x-1))dx = ∫((x2(x-1)+(x-1))/(x-1))dx
                       = ∫(((x2+1)(x-1))/(x-1))dx
                       = ∫(x2+1)dx
                       = (x3/3)+x+C                  
    Using, ∫xndx = (xn+1/(n+1))+C
    
  • Methods of Integration –



    1. Integration by Substitution :
      • Definition –The method of evaluating the integral by reducing it to standard form by proper substitution is called integration by substitution.
        If f(x) is a continuously differentiable function, then to evaluate the integral of the form

        ∫g(f(x))f(x)dx

        we substitute f(x)=t and f(x)’dx=dt.
        This reduces the integral to the form

        ∫g(t)dt
      • Examples :
        • Example 1.Evaluate the ∫e2x-3dx
        • Solution
          Let 2x-3=t => dx=dt/2
          ∫e2x-3dx = (∫etdx)/2
                  = (∫et)/2
                  = ((e2x-3)/2)+C
          
        • Example 2.Evaluate the ∫sin(ax+b)cos(ax+b)dx
        • Solution
          Let ax+b=t => dx=dt/a;
           ∫sin(ax+b)cos(ax+b)dx =  (∫sin(t)cos(t)dt)/a
                                 = (∫sin(2t)dt)/2a
                                 = -(cos(2t))/4a
                                 = (-cos(2ax+2b)/4a)+C
          
    2. Integration by Parts :
      • Theorem :If u and v are two functions of x, then
        ∫(uv)dx = u(∫vdx)-∫(u'∫vdx)dx

        where u is a first function of x and v is the second function of x

      • Choosing first function :
        We can choose first function as the function which comes first in the word ILATE where

        • I – stands for inverse trigonometric functions.
        • L – stands for logarithmic functions.
        • I – stands for algebraic functions.
        • I – stands for trigonometric functions.
        • I – stands for exponential functions.
      • Examples :
        • Example 1.Evaluate the ∫xsin(3x)dx
        • Solution
          Taking I= x and II = sin(3x)
          ∫xsin(3x)dx = x(∫sin(3x)dx)-∫((x)'∫sin(3x)dx)dx
                      = x(cos(3x)/(-3))-∫(cos(3x)/(-3))dx
                      = (xcos(3x)/(-3))+(cos(3x)/9)+C
          
        • Example 2.Evaluate the ∫xsec2xdx
        • Solution
          Taking I= x and II = sec2x
          ∫xsin(3x)dx = x(∫sec2xdx)-∫((x)'∫sec2xdx)dx
                      = (xtan(x))-∫(1*tan(x))dx
                      = xtan(x)+log|cos(x)|+C
          
    3. Integration by Partial Fractions :
      • Partial Fractions :
        If f(x) and g(x) are two polynomial functions, then f(x)/g(x) defines a rational function of x.
        If degree of f(x) < degree of g(x), then f(x)/g(x) is a proper rational function of x.
        If degree of f(x) > degree of g(x), then f(x)/g(x) is an improper rational function of x.
        If f(x)/g(x) is an improper rational function, we divide f(x) by g(x) so that the rational function can be represented as ∅(x) + (h(x)/g(x)).Now h(x)/g(x) is an proper rational function.
        Any proper rational function can be expressed as the sum of rational functions, each having a simple factor of g(x).Each such fraction is called partial fraction .
      • Cases in Partial Fractions :
        • Case 1.
          When g(x) = (x-a1)(x-a2)(x-a3)….(x-an), then we assume that

          f(x)/g(x) = (A1/(x-a1))+(A2/(x-a2))+(A3/(x-a3))+....(An/(x-an))
        • Case 2.
          When g(x) = (x-a)k(x-a1)(x-a2)(x-a3)
          ….(x-ar),
          then we assume that

          f(x)/g(x) = (A1/(x-a)1)+(A2/(x-a)2)+(A3/(x-a)3)
                            +....(Ak/(x-a)k)+(B1/(x-a1))+(B2/(x-a2))+(B3/(x-a3))
                            +....(Br/(x-ar))
      • Examples :
        • Example 1.∫(x-1)/((x+1)(x-2))dx
        • Solution
          Let (x-1)/((x+1)(x-2))= (A/(x+1))+(B/(x-2))
          =>    x-1 = A(x-2)+B(x+1)
          

          Putting x-2 = 0, we get

           B = 1/3 

          Putting x+1 = 0, we get

           A = 2/3  

          Substituting the values of A and B, we get

           
          (x-1)/((x+1)(x-2))= ((2/3)/(x+1))+((1/3)/(x-2))
          ∫((2/3)/(x+1))+((1/3)/(x-2))dx  
          = ((2/3)∫(x+1)dx)+((1/3)∫(x-2)dx)
          =  ((2/3)log|x+1|)+((1/3)log|x-2|)+C
          
        • Example 2.∫(cos(x))/((2+sin(x))(3+4sin(x)))dx
        • Solution
          Let I = ∫(cos(x))/((2+sin(x))(3+4sin(x)))dx 

          Putting sin(x) = t and cos(x)dx = dt, we get

           I = ∫dt/((2+t)(3+4t)) 
          Let 1/((2+t)(3+4t))= (A/(2+t))+(B/(3+4t))
          =>  1 = A(3+4t)+B(2+t) 

          Putting 3+4t = 0, we get

           B = 4/5 

          Putting 2+t = 0, we get

           A = -1/5 

          Substituting the values of A and B, we get

          1/((2+t)(3+4t))
          = ((-1/5)/(2+t))+((4/5)/(3+4t))
          I = (((-1/5)/(2+t))dt)+(((4/5)/(3+4t))dt)
           = ((-1/5)log|2+t|)+((1/5)log|3+4t|)+C
          = ((-1/5)log|2+sin(x)|)+((1/5)log|3+4sin(x)|)+C 
          


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