# Integral calculus - Integrals of rational functions

1. $\int adx =ax+c$
2. $\int xsx = {x^2 \over 2}+c$
3. $\int x^2dx= {x^3 \over 3}+c$
4. $\int x^pdx= {x^{p+1} \over p+1}+c, p \neq -1 \, .$
5. $\int (ax+b)^ndx= {(ax+b)^{n+1} \over a(n+1)}+c, n \neq -1.$
6. $\int {dx \over x}= In |x|+c$
7. $\int {dx \over ax+b}= {1 \over a}In |ax+b|+c$
8. $\int {ax+b \over cx+d}dx={a \over c}x+{bc-ad \over c^2}In |cx+d|+c$
9. $\int {dx \over (x+a)(x+b)}= {1 \over a-b}In \bigg| { x+b \over x+a} \bigg|+c, a \neq b.$
10. $\int {xdx \over a+bx}={1 \over b^2}(a+bx-a \:In |a+bx|)+c$
11. $\int {x^2dx \over a+bx}= {1 \over b^3} \bigg [ {1 \over 2} (a+bx)^2 -2a(a+bx)+a^2In |a+bx|\bigg ]+c$
12. $\int {dx \over x(a+bx)}= {1 \over a}in \bigg | {a+bx \over x } \bigg|+c$
13. $\int {dx \over x^2(a+bx)}= -{1 \over ax}+{b \over a^2}In \bigg | {a+bx \over x} \bigg |+c$
14. $\int {xdx \over (a+bx)^2}= {1 \over b^2} \bigg ( In |a+bx|+ {a \over a+bx} \bigg )+c$
15. $\int {x^2dx \over (a+bx)^2}= {1 \over b^3} \bigg ({a+bx-2a \: In |a+bx |-{a^2 \over a+bx}} \bigg )+c$
16. $\int {dx \over x(a+bx)^2}= {1 \over a(a+bx)}+ {1 \over a^2}In \bigg | {a+bx \over x} \bigg |+c$
17. $\int {dx \over x^2-1}= {1 \over 2}In \bigg |{ x-1 \over x+1} \bigg |+c$
18. $\int {dx \over 1-x^2}= {1 \over 2}In \bigg | {1+x \over 1-x} \bigg |+c$
19. $\int {dx \over a^2-x^2}= {1 \over 2a}In \bigg |{a+x \over a-x} \bigg |+c$
20. $\int {dx \over x^2-a^2}= {1 \over 2a}In \bigg |{x-1 \over x-a} \bigg |+c$
21. $\int {dx \over 1+x^2}=tan^{-1}x+c$
22. $\int {dx \over a^2+x^2}={1 \over a}tan^{-1}{x \over a}+c$
23. $\int {xdx \over x^2+a^2}={1 \over 2}In(x^2+a^2)+c$
24. $\int {dx \over a+bx^2}= {1 \over \sqrt {ab}}arctan \bigg (x\sqrt {b \over a } \bigg )+c, ab > 0.$
25. $\int {xdx \over a+bx^2}= {1 \over 2b}In \bigg |x^2+{a \over b} \bigg | +C$
26. $\int {dx \over x(a+bx^2)}= {1 \over 2a}In \bigg | {x^2 \over a+bx^2} \bigg | +c$
27. $\int {dx \over a^2-b^2x^2}= {1 \over 2ab }In \bigg | {a+bx \over a-bx} \bigg | +c$
28. $\int {dx \over ax^2+bx+c}= {1 \over \sqrt {b^2-4ac}}In \bigg | {2ax+b-\sqrt {b^2-4ac} \over 2ax+b+ \sqrt {b^2-4ac}} \bigg |+c,$
$b^2-4ac > 0.$
29. $\int {dx \over ax^2+bx+c}= {2 \over \sqrt {4ac-b^2}}arctan {2ax+b \over \sqrt {4ac- b^2}}+c,$
$b^2-4ac <0.$