Use partial fractions to solve the integral int (x^2-x+20)/(x^3+5x)dx

beatricalwu 2022-07-16 Answered
Use partial fractions to solve the integral
\int \frac{x^2-x+20}{x^3+5x}dx
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Answers (1)

grocbyntza
Answered 2022-07-17 Author has 25 answers
x 2 x + 20 x 3 + 5 x d x x 2 x + 20 x ( x 2 + 5 ) d x A x + B x + C x 2 + 5 = x 2 x + 20 x ( x 2 + 5 ) A ( x 2 + 5 ) + C x + 5 A = x 2 x + 20 A + B = 1 , C = 1 , 5 A = 20 B = 3 , A = 4 I = ( 4 x + 3 x 1 x 2 + 5 ) d x = 4 ln x 3 2 2 x d x x 2 + 5 d x x 2 + 5 = 4 ln x 3 2 ln ( x 2 + 5 ) 1 5 tan ( x 5 ) + C
[ in the 2nd term let  x 2 + 5 = v       x d x = d x / 2 ] d x x = ln x + C d x x 2 + a 2 = 1 a tan 1 ( x a ) I = 4 ln x 3 2 ln ( x 2 + 5 ) 1 5 tan 1 ( x 5 ) + C
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