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 .

Integration of quadratic equation in denominator



Click here to Watch more  Integration Videos

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 ) = 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 
Integration of quadratic equation in denominator
Now use the formula written below 

Integration of quadratic equation in denominator
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


HOW TO INTEGRATE INTEGRAL WITH SQUARE ROOT IN NUMERATOR



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 ) = 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 
HOW TO INTEGRATE INTEGRAL WITH SQUARE ROOT IN NUMERATOR

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.


TO CRACK  ANY COMPETITIVE EXAM VISIT THIS YOUTUBE CHANNEL FOR MATHS AND REASONING  TRICKS

EXAM CRACKER




Also Read     How to learn trigonometric identities easily
Share:

No comments:

Post a Comment

Recent Posts