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Evaluate these integrals...?
Hi there. I am having some problems answering these integrals and would appreciate the help. They each have to do with rational functions and/or integration.
1)Evaluate the integral: 6/(x+a)(x+b) dx (assume that a does not equal b)
2) (x^3-4x-7)/(x^2-x-6) dx from x=0 to x=1
3) 26/(x-1)(x^2+25) dx
4) 7dx/x(x^4+8)
5) 15arctanx/x^2 dx
2 Answers
- MechEng2030Lv 78 years ago
1.)
∫6/(x + a)(x + b) dx
= 6/(a - b) * ∫1/(x + b) - 1/(x + a) dx
= 6/(a - b) * ln|(x + b)/(x + a)| + C
2.)
∫(x^3 - 4x - 7)/(x - 3)(x + 2) dx from 0 to 1
= ∫(x)(x + 2)(x - 2)/(x - 3)(x + 2) dx - ∫7/(x - 3)(x + 2) dx
= ∫(x^2 - 2x)/(x - 3) dx - ∫7/(x - 3)(x + 2) dx
= ∫(x + 1 + 3/(x - 3)) dx - 7/5 * ∫1/(x - 3) - 1/(x + 2) dx
= (x^2)/2 + x + 3ln|x - 3| - 7/5*ln|(x - 3)/(x + 2)| eval. from 0 to 1
= 1/2 + 1 + 3ln(2) - 7/5*ln(2/3) - 3ln(3) + 7/5*ln(3/2)
= 3/2 + 3*ln(2/3) + 14/5*ln(3/2)
= 3/2 - 1/5*ln(3/2)
