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Integration shortcuts of f(x) when f(x) is as denominator has a derivative f΄(x) as nominator (Integration Tips)

This integration trick can be applied when f(x) is as denominator and its derivative is also exist as nominator.

In such case just take the natural log of f(x). Natrual log is represented as ln.



Let us we apply this integration rule on following examples.

  • Integration of 1/x
  •        /  
           | 1/x dx
           /
    

    (Here f(x) is x as denominator and its derivate is 1 is also exit as nominator. So apply the above rule. Just take the natural logarithm of f(x))

            = ln|x| + C
    
  • Integration of 1/xln(x)
  •        /  
           |    1 
           | -------- dx 
           |  x ln(x)
           /
    
              /   1/x
            = |  ----- . dx        
              |  ln(x)
              /
    

    (Here f(x) is ln(x) as denominator and its derivate is 1/x is also exist as nominator. So apply the above rule. Just take the natural logarithm of f(x))

            = ln(ln(x))  + C
    
  • Integration of 1/(1+x^2)tan^-1(x)
  •        /
           |        1   
           | ---------------- dx
           |       2      -1
           | (1 + x  ) tan  x 
           /
    
              /     1   
              |  --------   
              |        2
              |   1 + x  
           =  | ---------- dx
              |      -1
              |   tan  x 
              /
    

    (Here f(x) is tan-1 x as denominator and its derivate is 1/(1+x^2) is also exist as nominator. So apply the above rule. Just take the natural logarithm of f(x))

                  -1  
           = ln|tan x| + C
    
  • Integration of (1 + ln x)/xln(x)
  •        /
           |   1 + ln x
           |  ---------  dx
           |    x ln x 
           /
    


    First Method

    (Here f(x) is x ln x as denominator and its derivate is 1 + ln x is also exist as nominator. So apply the above rule. Just take the natural logarithm of f(x))

           = ln(x ln x) + C
    

    Second Method

              /
              |     1         ln x
            = | -------- + --------  dx
              |  x ln x     x ln x
              /
    
              /               /  
              |     1         |   ln x
            = | -------- dx + | ------- dx
              |  x ln x       |  x ln x
              /               / 
    
              /               /  
              |    1/x        |  1
            = | -------- dx + | --- dx
              |    ln x       |  x
              /               /
    

    (Here in first integral f(x) is ln x as denominator and its derivate is 1/x is also exist as nominator. So apply the above rule. Just take the natural logarithm of f(x))

    (Here in sencond integral f(x) is x as denominator and its derivate is 1 is also exist as nominator. So apply the above rule. Just take the natural logarithm of f(x))

            = ln (ln x) + ln x + C
            = ln (x ln x) + C          (By applying log property.)
    
    Integration Question

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  • Int. of x^n = x^(n+1)/n+1 is false when n = ?
  • Int. of x^0.1 = ?
    (x^0.1/0.1, x^0.2/0.2, x^1.1/1.1, x^2.2/2.2)
  • Int. of x^0.07 = ? (x^0.8/0.8, x^0.08/0.08, x^0.17/0.17, x^1.07/1.07)
  • What is integration of 1/2x . Int (-2x) ? (-x, -2x, -x/2, -x^2/2)


  • What is integration of (1+lnx)/xlnx ?
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  • What is integration of e^x = e^x/ ln e is true or false ?
  • What is integration of e^-x ?
  • What is integration of e^0 ?
  • What is integration of e^1 ?


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