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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. \(\int {dx \over ax^2+bx+c}= {2 \over \sqrt {4ac-b^2}}arctan {2ax+b \over \sqrt {4ac- b^2}}+c,\)
    \(b^2-4ac <0.\)