HOW TO INTEGRATE, INTEGRAL WITH SQUARE ROOT IN NUMERATOR

Let us find out the  Integration of quadratic polynomial  in denominator, integral of square root of polynomial, integrals with square roots in denominator, integration of linear by quadratic, integral with square root in numerator ,integration of square root formula .

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Let us take some examples to integrate such type of example.

Problem

1st of all making the co eff off  x²  positive and unity if it is not and then    rewrite  x² +4x -1 as sum/ difference 
of  (a + b )²  = a² +b² 土2ab  

and c²  as follows :- 

Now write the remaining term to equalise  x² + 4x -1,

Now writing the last term as square root of term and then multiplying the co eff of x²  with -ve sign which was earlier taken outside  .

Therefore (1) can be written as 

Now use the formula written below 

Here in this case 'x' will be replaced by 'x + 2' and 'a' will be replaced by √5

Where 'c' is called constant of integration.

This video will explains very easy and short method of such types of integration

Problem

1st of all making the co eff off  x²  positive and unity if it is not and then rewrite  2x² + x - 1 as sum / difference of 
 (a + b )²  = a² +b² 土2ab  

and c²  as follows :- 

2x² + x -1 =2[ x² +(1/2) x - (1/2)  ]
2x² + x -1 = 2[  (x )² + 2× (1/4)(x) +(1/4)² - (1/4)²  -1/2]

2x² + x -1 =  2[ {(x)² + 2 (1/4)(x) +(1/4)²}- {(1/4)²  +1/2}]

                =  2[{(x) + (1/4)}² - {1/16+1/2}]

                 = 2 [{(x) + (1/4)}² - {9/16}]

                 = 2[{(x) + (1/4)}² -  {3/4}² ]

Now using the Result 

Now replacing x by x + (1/4) and 'a' by 3/4 ,
The integration of this Integrand can be written as follows

Where 'c' is called constant of integration.

Now multiplying with 2 all the terms and After simplification we shall have 

In this post we discussed Integration of quadratic polynomial  in denominator,integral of   square root of polynomial,integrals with square roots in denominator, integration of linear by quadratic, integral with square root in numerator ,integration of square root formula,integration of square root formula,integral of root 2, anti derivative rules for square roots Please like it and  share this post with yours friends.

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