I need to compute the integral ∫ 2 x ( x 2 + x +...

Janet Forbes

Janet Forbes

Answered

2022-07-04

I need to compute the integral 2 x ( x 2 + x + 1 ) 2 d x. I tried using the integration of a rational function technique, with A x + B x 2 + x + 1 + C x + D ( x 2 + x + 1 ) 2 , but this simply returned C = 2 and A , B , D = 0, so it doesn't really change anything.
I also tried using a u substitution, setting u = x 2 + x + 1. This made the numerator 2 x = d u d x 1, but I'm not really sure if I can do that/how to solve an integral with a derivative as a part of it.
How would I go about solving this?

Answer & Explanation

1s1zubv

1s1zubv

Expert

2022-07-05Added 17 answers

2 x ( x 2 + x + 1 ) 2
2 x ( ( x + 1 2 ) 2 + 3 4 ) 2
Apply u-substitution: u = x + 1 2
2 8 ( 2 u 1 ) ( 4 u 2 + 3 ) 2 d u
2 ( 8 ) 2 u 1 ( 4 u 2 + 3 ) 2 d u
Apply the Sum Rule
2 ( 8 ) ( 2 u ( 4 u 2 + 3 ) 2 d u 1 ( 4 u 2 + 3 ) 2 d u )
Now,
2 u ( 4 u 2 + 3 ) 2 = 1 4 ( 4 u 2 + 3 )
Now,
1 ( 4 u 2 + 3 ) 2 d u = 1 12 3 ( a r c t a n ( 2 3 u ) + 1 2 s i n ( 2 a r c t a n ( 2 3 u ) ) )
= 2 ( 8 ) ( 1 4 ( 4 u 2 + 3 ) 1 12 3 ( a r c t a n ( 2 3 u ) + 1 2 s i n ( 2 a r c t a n ( 2 3 u ) ) )
After doing small calculations and substituting u = x + 1 2 ,
2 x x 2 + x + 1 d x = 16 ( 1 4 ( 4 x 2 + 4 x + 4 ) 1 24 3 ( 2 a r c t a n ( 2 x + 1 3 ) + s i n ( 2 a r c t a n ( 2 x + 1 3 ) ) ) ) + C
civilnogwu

civilnogwu

Expert

2022-07-06Added 6 answers

Apply integration by parts to
1 x 2 + x + 1 d x
and extract your desired integral from the result.

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