I am trying to calculate the integral <msubsup> &#x222B;<!-- ∫ --> <mrow class="MJX-T

crossoverman9b

crossoverman9b

Answered question

2022-06-06

I am trying to calculate the integral
a + x b 2 + ( a + x ) 2 1 1 + c ( a x ) 2 d x
where { a , b , c } R . I have looked in a table of integrals for rational functions, but with no luck. Is there a smart trick I can utilize?

Answer & Explanation

podesect

podesect

Beginner2022-06-07Added 20 answers

Hint: Use partial fraction to write the integral as
a + x b 2 + ( a + x ) 2 1 1 + c ( a x ) 2 = A + B x b 2 + ( a + x ) 2 + C + D x 1 + c ( a x ) 2 ,
then you can write it as
A b 2 + ( a + x ) 2 + C 1 + c ( a x ) 2 + B x b 2 + ( a + x ) 2 + D x 1 + c ( a x ) 2 ,
and finally make the change of variables x a = t and x + a = t and things will be easy.
Kassandra Ross

Kassandra Ross

Beginner2022-06-08Added 6 answers

Potential Hint:
a ( b 2 + ( a + x ) 2 ) ( 1 + c ( a x ) 2 ) x ( b 2 + ( a + x ) 2 ) ( 1 + c ( a x ) 2 ) d x
The first part can be solved using a chain rule like method. The second half can be solved using integration by parts with u = x i.e. u = 1
Hope this helps!

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