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Evaluate: int(a x^2+b x+c)/((x-a)(x-b)(x...

Evaluate: `int(a x^2+b x+c)/((x-a)(x-b)(x-c))\ dx ,\ \ w h e r e\ a ,\ b ,\ c` are distinct.

Text Solution

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We have,
`I=int frac{a x^{2}+b x+c}{(x-a)(x-b)(x-c)} d x `
`text { Let } frac{a x^{2}+b x+c}{(x-a)(x-b)(x-c)}=frac{A}{x-a}+frac{B}{x-b}+frac{C}{x-c} `
`Rightarrow a x^{2}+b x+c=A(x-b)(x-c)+B(x-c)(x-a)+C(x-a)(x-b) `
`Rightarrow a x^{2}+b x+c=A[x^{2}-(b+c) x+b c]+B[x^{2}-(c+a) x+c a]+C[x^{2}-(a+b) x+a b] `
`Rightarrow a x^{2}+b x+c=(A+B+C) x^{2}-[A(b+c)+B(c+a)+C(a+b)] x+A b c+B c a+C a b`
...
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