7.4 Integration of Rational Functions by the Method of Partial Fractions
1. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-1.html
-x/(x^2 - 2*x + 1) + 2*log(x - 1) - log(x)
2. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-2.html
1/2*log(x-1)+1/2*log(x+1)-log(x)
3. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-3.html
2/3*log(x - 5) + 1/3*log(x + 1)
4. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-4.html
-4/(x^2 - 4) + 3*log(x - 2) + 2*log(x + 2)
5. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-5.html
2/3*arctan(1/2*x) - 1/3*arctan(x)
6. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-6.html
1/2*x^2 + 2*x + 3/2*log(x - 1) - 1/2*log(x + 1)
7. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-7.html
1/2*log(x^2 + 1) + log(x)
8. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-8.html
1/3*sqrt(3)*arctan(1/3*(2*x-1)*sqrt(3))+1/3*log(x+1)-1/6*log(x^2-x+1)
9. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-9.html
-5/(x - 3) + 3*log(x - 3)
10. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-10.html
-1/6*sqrt(3)*arctan(1/3*(2*x-1)*sqrt(3))-1/6*sqrt(3)*arctan(1/3*(2*x+1)*sqrt(3))+1/6*log(x-1)-1/6*log(x+1)
+1/12*log(x^2-x+1)-1/12*log(x^2+x+1)
11. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-11.html
-1/8*sqrt(2)*log(x^2-sqrt(2)*x+1)+1/8*sqrt(2)*log(x^2+sqrt(2)*x+1)
+1/4*sqrt(2)*arctan(1/2*(2*x-sqrt(2))*sqrt(2))+1/4*sqrt(2)*arctan(1/2*(2*x+sqrt(2))*sqrt(2))
12. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-12.html
3/5*arctan(1/3*x) - 2/5*arctan(1/2*x)
13. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-13.html
-1/2*x/(x^2 + 1) + 1/2*arctan(x)
14. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-14.html
-1/4*(x-1)/(x^2+1)+1/4*log(x-1)-1/8*log(x^2+1)-1/2*arctan(x)
15. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-15.html
-(14*x^2 - 34*x + 19)/(x^3 - 4*x^2 + 5*x - 2) - 14*log(x - 2) + 14*log(x - 1)
16. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-16.html
-4*log(x + 1) + 13/2*log(2*x + 3)
17. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-17.html
sqrt(x^2 + 9) - 3*arcsinh(3/abs(x))
18. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-18.html
-1/2*(12*x^3 - 18*x^2 + 4*x + 1)/(x^4 - 2*x^3 + x^2) - 6*log(x - 1)+ 6*log(x)
19. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-19.html
A^3*log(x) + 1/6*(2*x^(3*n) + 18*x^n*A^2 + 9*A*x^(2*n))/n
.
.
.
20. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-20.html
-1/9*pi*sqrt(3) + 1/3*log(2) + 1/2
21. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-21.html
1/2*log(x^2 + 4) + log(x) - 1/2*arctan(1/2*x)
22. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-22.html
1/2*x/(x^2 + 1) + 1/2*arctan(x)
23. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-23.html
1/3*sqrt(3)*log((sqrt(x + 2) - sqrt(3))/(sqrt(x + 2) + sqrt(3)))1/3*sqrt(3)*log((sqrt(x + 2) - sqrt(3))/(sqrt(x + 2) + sqrt(3)))
24. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-24.html
4*sqrt(-(x + 2)/(x - 2))/((x + 2)/(x - 2) - 1) + 4*arctan(sqrt(-(x +2)/(x - 2)))
25. .
http://matrix.skku.ac.kr/cal-lab/cal-7-4-25.html
-6*x^(1/6) - 3*log(x^(1/6) - 1) + 3*log(x^(1/6) + 1)
26..
.
27. . [Hint: Substitute ]
http://matrix.skku.ac.kr/cal-lab/cal-7-4-27.html
-1/2*sqrt(2)*arcsinh(x/abs(x) + 2/abs(x))