Help with Calculate integral Find <msubsup> &#x222B;<!-- ∫ --> 0 a </msubsup>

Feinsn

Feinsn

Answered question

2022-06-14

Help with Calculate integral
Find 0 a 3 x 2 a x ( x 2 a ) ( x 2 + a 2 ) d x I tried using partial fractions and the substitution u = a x but I haven't made any real progress. Please help.

Answer & Explanation

mar1nerne

mar1nerne

Beginner2022-06-15Added 20 answers

Use partial fraction
3 x 2 a x ( x 2 a ) ( x 2 + a 2 ) = A ( x 2 a ) + B x + C ( x 2 + a 2 )
Now find A,B,C
A = 2 B = 1 C = a 3 x 2 a x ( x 2 a ) ( x 2 + a 2 ) = 2 ( x 2 a ) + x + a ( x 2 + a 2 ) = 3 x 2 a x ( x 2 a ) ( x 2 + a 2 ) = 2 ( x 2 a ) + x ( x 2 + a 2 ) + a ( x 2 + a 2 )
So
0 a 3 x 2 a x ( x 2 a ) ( x 2 + a 2 ) d x = 0 a ( 2 ( x 2 a ) + x ( x 2 + a 2 ) + a ( x 2 + a 2 ) ) d x = ( 2 ln | x 2 a | + 1 2 ln ( x 2 + a 2 ) + tan 1 ( x a ) ) | 0 a .
arridsd9

arridsd9

Beginner2022-06-16Added 12 answers

Hint: Use the formular:
3 x 2 a x ( x 2 a ) ( x 2 + a 2 ) = A ( x 2 a ) + B x + C ( x 2 + a 2 )
to resolve 3 x 2 a x ( x 2 a ) ( x 2 + a 2 ) into partial fraction by finding the values of A, B and C. Hope you can do that?
Then
0 a 3 x 2 a x ( x 2 a ) ( x 2 + a 2 ) d x = 0 a A ( x 2 a ) d x + 0 a B x + C ( x 2 + a 2 ) d x .

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