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Integration shortcuts of [f(x)]^n also has derivative f'(x) (Integration Tips)

This integration trick can be applied when f(x) has a power n (here n ≠ 1) And derivate of f(x) is also persent of front of [f(x)]^n. Keep in mind always that n should not be equal to -1

In such case just take the f(x), take its power with plus 1 as power and divide by new power.



Let us we apply this integration rule on following examples.

  • Integration of x^2 or Integration of x raise to power 2
  •        /  2
           | x dx
           /
    
              /  2
            = | x . 1 dx        
              /
    

    (Here f(x) is x and 2 is its power, derivative of x is 1 which is also exist in front of x. So apply the above rule. Just take f(x), take its power with plus 1 as power and divide by new power.)

                 2 + 1    
                x
            = -------  + C
               2 + 1
    
                 3
                x
            = -----  + C
                3
    
    
  • Integration of 1
  •        /  
           | 1 dx
           /
    
              /  0
            = | x . dx        
              /
    
              /  0
            = | x . 1 dx        
              /
    

    (Here f(x) is x and 0 is its power, derivative of x is 1 which is also exist in front of x. So apply the above rule. Just take f(x), take its power with plus 1 as power and divide by new power.)

                 0 + 1    
                x
            = -------  + C
               0 + 1
    
                 1
                x
            = -----  + C
                1
    
            = x + C
    
  • Integration of (ln(x))/x
  •        /
           | ln(x)   
           | ----- dx
           |   x
           /
    
              /
              |          1   
            = | ln(x) . --- dx
              |          x
              /
    

    (Here f(x) is ln(x) and 1 is its power, derivative of ln(x) is 1/x which is also exist in front of ln(x).So apply the above integration rule. Just take f(x), take its power with plus 1 as power and divide by new power.)

                    1 + 1    
              (ln(x))
            = -------  + C
               1 + 1
    
                     2
              (ln x)
            = -----  + C
                2
    
  • Integration of (2x +3)^(1/2)
  •        /
           |        1/2    
           | (2x + 3) dx
           /
    
                 /
               1 |        1/2    
            = ---| (2x + 3)   . 2 dx
               2 | 
                 /
    

    (Here f(x) is 2x + 3 and 1/2 is its power, its derivative was not exit so we try to make its derivative by multiply and divided by 2. Now derivative of 2x + 3 is 2 which is in front of 2x + 3. So apply the above rule. Just take f(x), take its power with plus 1 as power and divide by new power.)

                          1                    
                         ---  + 1
                          2    
                1  (2x + 3)   
            =  --- --------  + C
                2   1 
                   --- + 1
                    2 
    
                          3                    
                         --- 
                          2    
                1  (2x + 3)   
            =  --- --------  + C 
                2     3 
                     ---
                      2 
                         
                       3/2    
               (2x + 3)   
            =  --------  
                  3 
    
    
    Integration Question

  • What is use of integration in our daily life ?
  • What is integration of ZERO ?
  • What is integration of C (constant) ?


  • What is integration of 1 ?
  • The power formula of int. use when ?
  • 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 ?
  • What is integration of e^x ?
  • 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 ?


  • Integration Shortcuts (Solve in seconds)

  • Integration shortcuts of [f(x)]^n also has derivative f'(x)
  • Integration shortcuts of f(x) when f(x) is as denominator and has a derivative f΄(x) as nominator


  • Integration Worksheets with solution

  • Integration Worksheets, Integration basics, Integration by parts, Integration by subtitutions with solution and downloadable.
  • CALCULAS Questions


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