Evaluate the integral int (1)/(x^3-1)dx

suchonosdy 2022-07-14 Answered
Evaluate the integral
1 x 3 1 d x
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Answers (1)

Monica Dennis
Answered 2022-07-15 Author has 13 answers
Evaluate the integral
1 x 3 1 d x = 1 ( x 1 ) ( x 2 + x + 1 ) d x = A x 1 d x + B x + C x 2 + x + 1 d x
Solving 1 = A ( x 2 + x + 1 ) + ( B x + C ) ( x 1 ) 1 = A x 2 + A x + A + B x 2 B x + C x C 1 = ( A + B ) x 2 + ( A B + C ) x + A C 0 = A + B B = A 0 = A B + C 1 = A C C = A 1
0 = A ( A ) + A 1 0 = 3 A 1 A = 1 3 B = 1 3 ,       C = 1 3 1 = 2 3 = 1 3 d x x 1 + 1 3 x 2 3 x 2 + x + 1 d x = 1 3 d x x 1 1 3 x + 2 x 2 + x + 1 d x = 1 3 ln | x 1 | 1 3 x + 2 ( x + 1 2 ) 2 + 3 4 d x u = x + 1 2 d u = d x = 1 3 ln | x 1 | 1 3 u 1 2 + 2 u 2 + 3 4 d u = 1 3 ln | x 1 | 1 3 u + 3 2 u 2 + 3 4 d u = 1 3 ln | x 1 | 1 3 u u 2 + 3 4 1 3 3 2 u 2 + 3 4 d u
= 1 3 ln | x 1 | 1 6 ln | w | 1 2 ( 1 3 2 a r c t a g ( 2 3 u ) ) + c = 1 3 ln | x 1 | 1 6 ln | ( x + 1 2 ) 2 + 3 4 | 3 3 a r c t a g ( 2 x + 1 3 ) + C
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