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Integral Calculus Tables


Rules

  1. \(\int c \cdot f \left( x \right) dx = c \cdot \int f \left( x \right) dx\)
  2. \(\int \left( f \left( x \right) \pm g \left( x \right) \right) dx = \int f \left( x \right) dx \pm \int g \left( x \right) dx\)
  3. \(\int f \left( g \left( x \right) \right) g^{\prime} \left( x \right) dx = \int f \left( u \right) du = F \left( u \right) + C = F\left(g\left( x \right)\right) + C \; ; \; F^{\prime} = f\)
  4. \(\int u dv = uv - \int v du\)

Rational functions

  1. \(\int c\ dx = cx + C\)
  2. \(\int x\ dx = \frac{x^{2}}{2} + C\)
  3. \(\int x^{n} dx = \frac{nx^{n + 1}}{n + 1} + C ,\ n \neq -1\)
  4. \(\int \left( ax^{c} + b \right)^{n}dx = \frac{1}{a} \int \left( ax^{c} + b \right)^{n} a dx = \frac{1}{a} \frac{\left( ax^{c} + b \right)^{n+1}}{\left( n + 1 \right)c x} + C,\ n \neq -1\)
  5. \(\int \left( ax + b \right)^{n}dx = \frac{1}{a} \int \left( ax + b \right)^{n} a dx = \frac{1}{a} \frac{\left( ax + b \right)^{n+1}}{ n + 1 } + C,\ n \neq -1\)
  6. \(\int \frac{dx}{ax + b} = \frac{1}{a} \int \left( ax + b \right)^{-1} a dx = \frac{1}{a} \ln\vert ax + b\vert + C\)
  7. \(\int x \left( ax + b \right)^{n}dx = \frac{a\left( n + 1 \right) x - b}{a^{2} \left( n + 1 \right)\left( n + 2 \right)} \left( ax + b \right)^{n+1}+ C,\ n \notin \left\lbrace -1,-2 \right\rbrace\)
  8. \(\int \frac{dx}{x} = \ln \vert x \vert + C\)
  9. \(\int \frac{dx}{ax + b} = \frac{1}{a} \int \left( ax + b \right)^{-1} a dx = \frac{1}{a}\ln \vert ax + b \vert + C\)
  10. \(\int \frac{x\ dx}{ax + b} = \frac{x}{a} - \frac{b}{a^{2}}\ln \vert ax + b \vert + C\)
  11. \(\int \frac{x\ dx}{\left(ax + b\right)^{2}} = \frac{b}{a^{2}\left( ax + b \right)} - \frac{1}{a^{2}}\ln \vert ax + b \vert + C\)
  12. \(\int \frac{x\ dx}{\left(ax + b\right)^{n}} = \frac{a \left( 1 - n \right)x - b}{a^{2} \left( n - 1 \right) \left( n - 2 \right)\left( ax + b \right)^{n-1}} + C,\ n \notin \left\lbrace -1,-2 \right\rbrace\)
  13. \(\int \frac{x^{2}dx}{ax + b} = \frac{1}{a^{3}} \left( \frac{\left( ax + b \right)^{2}}{2} - 2b\left( ax + b \right) + b^{2} \ln \vert ax + b \vert\right)+ C\)
  14. \(\int \frac{x^{2}dx}{\left(ax + b\right)^{2}} = \frac{1}{a^{3}} \left( ax + b - 2b \ln \vert ax + b \vert - \frac{b^{2}}{ax + b} \right) + C\)
  15. \(\int \frac{x^{2}dx}{\left(ax + b\right)^{3}} = \frac{1}{a^{3}} \left( \ln \vert ax + b \vert + \frac{2b}{ax + b} - \frac{b^{2}}{2 \left(ax + b\right)^{2}} \right) + C\)
  16. \(\int \frac{x^{2}dx}{\left(ax + b\right)^{n}} = \frac{1}{a^{3}} \left( - \frac{1}{\left( n - 3 \right) \left( ax + b \right)^{n-3}} + \frac{2b}{\left( n - 2 \right) \left( a + b \right)^{n-2}} - \frac{b^{2}}{\left( n-1 \right)\left( ax + b \right)^{n-1}} \right) + C,\ n \notin \left\lbrace -1,-2,-3 \right\rbrace\)
  17. \(\int \frac{dx}{x\left(ax + b\right)} = - \frac{1}{b} \ln \left\vert \frac{ax + b}{x} \right\vert + C\)
  18. \(\int \frac{dx}{x^{2}\left(ax + b\right)} = - \frac{1}{bx} + \frac{a}{b^{2}}\ln \left\vert \frac{ax + b}{x} \right\vert + C\)
  19. \(\int \frac{dx}{x^{2}\left(ax + b\right)^{2}} = - a\left(\frac{1}{b^{2}\left(ax + b\right)} + \frac{1}{ab^{2}x} - \frac{2}{b^{3}}\ln \left\vert \frac{ax + b}{x} \right\vert\right) + C\)
  20. \(\int \frac{dx}{x^{2} + a^{2}} = \frac{1}{a} \arctan \frac{x}{a}+ C\)
  21. \(\int \frac{dx}{x^{2} - a^{2}} = -\frac{1}{a} \operatorname{artanh} \frac{x}{a} + C= \frac{1}{2a} \ln\frac{a-x}{a+x} + C,\ \vert x \vert < \vert a \vert\)
  22. \(\int \frac{dx}{x^{2} - a^{2}} = -\frac{1}{a} \operatorname{arcoth} \frac{x}{a} + C= \frac{1}{2a} \ln\frac{x-a}{x+a} + C,\ \vert x \vert > \vert a \vert\)
  23. \(\int \frac{dx}{ax^{2} + bx + c} = \frac{2}{\sqrt{4ac-b^{2}}} \arctan\frac{2ax + b}{\sqrt{4ac - b^{2}}} + C,\ 4ac - b^{2} > 0\)
  24. \(\int \frac{dx}{ax^{2} + bx + c} = \frac{2}{\sqrt{b^{2}-4ac}} \operatorname{artanh} \frac{2ax + b}{\sqrt{b^{2} - 4ac}} = \frac{1}{\sqrt{b^{2} - 4ac}} \ln \left\vert \frac{2ax + b - \sqrt{b^{2}-4ac}}{2ax + b + \sqrt{b^{2}-4ac}} \right\vert + C,\ 4ac - b^{2} < 0\)
  25. \(\int \frac{xdx}{ax^{2} + bx + c} = \frac{1}{2a} \ln \vert ax^{2} + bx + c \vert - \frac{b}{2a} \int \frac{dx}{ax^{2} + bx +c}\)
  26. \(\int \frac{mx + n}{ax^{2} + bx + c}\ dx = \frac{m}{2a} \ln \vert ax^{2} + bx + c \vert + \frac{2an - bm}{a\sqrt{4ac-b^2}} \arctan \frac{2ax + b}{\sqrt{4ac - b^{2}}} + C,\ 4ac - b^{2} > 0\)
  27. \(\int \frac{mx + n}{ax^{2} + bx + c}\ dx = \frac{m}{2a} \ln \vert ax^{2} + bx + c \vert + \frac{2an - bm}{a\sqrt{b^2 -4ac}} \operatorname{artanh} \frac{2ax + b}{\sqrt{b^{2} - 4ac}} + C,\ 4ac - b^{2} < 0\)
  28. \(\int \frac{dx}{\left(ax^{2} + bx + c\right)^{n}} = \frac{2ax + b}{\left( n-1 \right)\left( 4ac - b^{2} \right)\left( ax^{2} + bx + c \right)^{n-1}} + \frac{2a\left( 2n - 3 \right)}{\left( n-1 \right)\left( 4ac -b^{2} \right)} \int \frac{dx}{\left( ax^{2} + bx + c \right)^{n-1}} + C\)
  29. \(\int \frac{x\ dx}{\left(ax^{2} + bx + c\right)^{n}} = \frac{bx + 2c}{\left( n-1 \right)\left( 4ac - b^{2} \right)\left( ax^{2} + bx + c \right)^{n-1}} + \frac{b\left( 2n - 3 \right)}{\left( n-1 \right)\left( 4ac -b^{2} \right)} \int \frac{dx}{\left( ax^{2} + bx + c \right)^{n-1}} + C\)
  30. \(\int \frac{dx}{x \left(ax^{2} + bx + c\right)} = \frac{1}{2c} \ln \left\vert \frac{x^{2}}{ax^{2} + bx + c} \right\vert - \frac{b}{2c} \int \frac{dx}{ax^{2} + bx + c} + C\)

Irrational functions

Integrals involving \( r = \sqrt{x^{2} + a^{2}} \)

  1. \(\int r\ dx = \frac{1}{2} \left( xr + a \ln\left( x + r \right) \right) + C\)
  2. \(\int r^{3} dx = \frac{1}{4} xr^{3} + \frac{1}{8} a^{2}xr + \frac{3}{8} a^{4} \ln\left( x + r \right) + C\)
  3. \(\int r^{5} dx = \frac{1}{6} xr^{5} + \frac{5}{24} a^{2}xr^{3} + \frac{5}{16} a^{4}xr + \frac{5}{16} \ln\left( x + r \right) + C\)
  4. \(\int xr\ dx = \frac{r^{3}}{3} + C\)
  5. \(\int xr^{3} dx = \frac{r^{5}}{5} + C\)
  6. \(\int xr^{2n + 1} dx = \frac{r^{2n + 3}}{2n + 3} + C\)
  7. \(\int x^{2}r dx = \frac{xr^{3}}{4} - \frac{a^{2}xr}{8} - \frac{a^{4}}{8} \ln \left( x + r \right) + C\)
  8. \(\int x^{2}r^{3} dx = \frac{xr^{5}}{6} - \frac{a^{2}xr^{3}}{24} - \frac{a^{4}xr}{16} - \frac{a^{6}}{16} \ln \left( x + r \right) + C\)
  9. \(\int x^{3}r dx = \frac{r^{5}}{5} - \frac{a^{2}r^{3}}{3} + C\)
  10. \(\int x^{3}r^{3} dx = \frac{r^{7}}{7} - \frac{a^{2}r^{5}}{5} + C\)
  11. \(\int x^{3}r^{2n + 1} dx = \frac{r^{2n + 5}}{2n + 5} - \frac{a^{2}r^{2n + 3}}{2n + 3} + C\)
  12. \(\int x^{4}r\ dx = \frac{x^{3}r^{3}}{6} - \frac{a^{2}xr^{3}}{8} + \frac{a^{4}xr}{16} + \frac{a^{6}}{16} \ln \left( x + r \right) + C\)
  13. \(\int x^{4}r^{3} dx = \frac{x^{3}r^{5}}{8} - \frac{a^{2}xr^{5}}{16} + \frac{a^{4}xr^{3}}{64} + \frac{3a^{6}xr}{128} + \frac{3a^{8}}{128} \ln \left( x + r \right) + C\)
  14. \(\int x^{5}r\ dx = \frac{r^{7}}{7} - \frac{2a^{2}r^{5}}{5} + \frac{a^{4}r^{3}}{3} + C\)
  15. \(\int x^{5}r^{3} dx = \frac{r^{9}}{9} - \frac{2a^{2}r^{7}}{7} + \frac{a^{4}r^{5}}{5} + C\)
  16. \(\int x^{5}r^{2n + 1} dx = \frac{r^{2n + 7}}{2n + 7} - \frac{2a^{2}r^{2n + 5}}{2n + 5} + \frac{a^{4}r^{2n + 3}}{2n + 3} + C\)
  17. \(\int \frac{r\ dx}{x} = r - a \ln \left\vert \frac{a + r}{x}\right\vert + C = r - a \sinh^{-1} \frac{a}{x} + C\)
  18. \(\int \frac{r^{3}\ dx}{x} = \frac{r^{3}}{3} + a^{2}r - a^{3} \ln \left\vert \frac{a + r}{x}\right\vert + C\)
  19. \(\int \frac{r^{5}\ dx}{x} = \frac{r^{5}}{5} + \frac{a^{2}r^{3}}{3} + a^{4}r - a^{5} \ln \left\vert \frac{a + r}{x}\right\vert + C\)
  20. \(\int \frac{r^{7}\ dx}{x} = \frac{r^{7}}{7} + \frac{a^{2}r^{5}}{5} + \frac{a^{4}r^{3}}{3} + a^{6}r - a^{7} \ln \left\vert \frac{a + r}{x}\right\vert + C\)
  21. \(\int \frac{dx}{r} = \sinh^{-1} \frac{x}{a} + C= \ln \vert x + r \vert + C\)
  22. \(\int \frac{dx}{r^{3}} = \frac{x}{a^{2}r} + C\)
  23. \(\int \frac{x\ dx}{r^{3}} = -\frac{1}{r} + C\)
  24. \(\int \frac{x^{2}dx}{r} = \frac{x}{2}r - \frac{a^{2}}{2} \sinh^{-1} \frac{x}{a} + C = \frac{x}{2}r - \frac{a^{2}}{2} \ln \left\vert x + r \right\vert + C\)
  25. \(\int \frac{dx}{xr} = -\frac{1}{a} \sinh^{-1} \frac{a}{x} + C = -\frac{1}{a} \ln \left\vert \frac{a + r}{x} \right\vert + C\)

Integrals involving \( s = \sqrt{x^{2} - a^{2}} \)

  1. \(\int x s\ dx= \frac{1}{3} s^{3} + C\)
  2. \(\int \frac{s\ dx}{x} = s-a\cos^{-1} \left\vert \frac{a}{x} \right\vert + C\)
  3. \(\int \frac{dx}{xs} = \frac{1}{a}\cos^{-1} \left\vert \frac{a}{x} \right\vert + C = \frac{1}{a} \operatorname{sgn} x \cos^{-1}\frac{a}{x} + C\)
  4. \(\int \frac{dx}{s} = \ln \left\vert \frac{x+s}{a} \right\vert + C\)
  5. \(\int \frac{x\ dx}{s} = s + C\)
  6. \(\int \frac{x\ dx}{s^{3}} = -\frac{1}{s} + C\)
  7. \(\int \frac{x\ dx}{s^{5}} = -\frac{1}{3s^{3}} + C\)
  8. \(\int \frac{x\ dx}{s^{7}} = -\frac{1}{5s^{5}} + C\)
  9. \(\int \frac{x\ dx}{s^{2n + 1}} = - \frac{1}{(2n - 1)s^{2n - 1}}\cdots\)
  10. \(\int \frac{x^{2m}\ dx}{s^{2n+1}} = -\frac{1}{2n-1} \frac{x^{2m-1}}{s^{2n-1}}+ \frac{2m-1}{2n-1} \int\frac{x^{2m-2}dx}{s^{2n-1}} + C\)
  11. \(\int \frac{x^{2}\ dx}{s} = \frac{xs}{2} + \frac{a^{2}}{s}\ln \left\vert \frac{x + s}{a} \right\vert + C\)
  12. \(\int \frac{x^{2}dx}{s^{3}} = -\frac{x}{s} + \ln \left\vert \frac{x + s}{a} \right\vert + C\)
  13. \(\int \frac{x^{4}\ dx}{s} = \frac{x^{3}s}{4}+\frac{3}{8}a^{2}xs+\frac{3}{8}a^{4}\ln \left\vert\frac{x + s}{a} \right\vert + C\)
  14. \(\int \frac{x^{4}\ dx}{s^{3}} = \frac{xs}{2}-\frac{a^{2}x}{s}+\frac{3}{2}a^{2}\ln \left\vert\frac{x + s}{a} \right\vert + C\)
  15. \(\int \frac{x^{4}\ dx}{s^{5}} = - \frac{x}{s} - \frac{1}{3}\frac{x^{3}}{s^{3}} + \ln \left\vert\frac{x + s}{a} \right\vert + C\)
  16. \(\int \frac{x^{2m}\ dx}{s^{2n + 1}} = \left (-1 \right)^{n-m} \frac{1}{a^{2(n-m)}}\displaystyle\sum_{i=0}^{n-m-1}\frac{1}{2(m+i)+1} \binom{n-m-1}{i}\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}},\\ (n>m \geq 0)\)
  17. \(\int \frac{dx}{s^{3}} = - \frac{1}{a^{2}}\frac{x}{s} + C\)
  18. \(\int \frac{dx}{s^{5}} = \frac{1}{a^{4}} \left\lbrack \frac{x}{s}-\frac{1}{3}\frac{x^{3}}{s^{3}} \right\rbrack + C\)
  19. \(\int \frac{dx}{s^{7}} = -\frac{1}{a^{6}}\left\lbrack\frac{x}{s}-\frac{2}{3}\frac{x^{3}}{s^{3}}+\frac{1}{5}\frac{x^{5}}{s^{5}} \right\rbrack + C\)
  20. \(\int \frac{dx}{s^{9}} = \frac{1}{a^{8}}\left\lbrack \frac{x}{s}- \frac{3}{3}\frac{x^{3}}{s^{3}}+\frac{3}{5}\frac{x^{5}}{s^{5}}-\frac{1}{7}\frac{x^{7}}{s^{7}}\right\rbrack + C\)
  21. \(\int \frac{x^{2}\ dx}{s^{5}} = -\frac{1}{a^{2}}\frac{x^{3}}{3s^{3}} + C\)
  22. \(\int \frac{x^{2}\ dx}{s^{7}} = \frac{1}{a^{4}}\left\lbrack \frac{1}{3}\frac{x^{3}}{s^{3}}-\frac{1}{5}\frac{x^{5}}{s^{5}}\right\rbrack + C\)
  23. \(\int \frac{x^{2}dx}{s^{9}} = -\frac{1}{a^{6}}\left\lbrack \frac{1}{3}\frac{x^{3}}{s^{3}}- \frac{2}{5}\frac{x^{5}}{s^{5}}+\frac{1}{7}\frac{x^{7}}{s^{7}} \right\rbrack + C\)
  24. \(\int s\ dx = \frac{1}{2}\left( xt - \operatorname{sgn} x \cosh^{-1}{\left\vert\frac{x}{a}\right\vert}\right) + C\)
  25. \(\int f \left( \frac{h(x)}{g(x)\ s} \right) dx = \operatorname{sgn} x\ Fx + C\)

Integrals involving \( t = \sqrt{a^{2} - x^{2}} \)

  1. \(\int t\ dx = \frac{1}{2}\left( xt + a^{2} \sin^{-1}\left( \frac{x}{a} \right) \right) + C\)
  2. \(\int x\ t\ dx = -\frac{1}{3} t^{3} + C\)
  3. \(\int \frac{t\ dx}{x} = t - a \ln \left\vert \frac{a + t}{x} \right\vert + C\)
  4. \(\int \frac{dx}{t} = \sin^{-1}\left(\frac{x}{a}\right) + C\)
  5. \(\int \frac{x^{2}\ dx}{t} = -\frac{x}{2}t + \frac{a^{2}}{2} \sin^{-1}\left(\frac{x}{a}\right) + C\)

Integrals involving \( R = \sqrt{ax^{2} + bx + c} \)

  1. \(\int \frac{dx}{R} = \frac{1}{\sqrt{a}} \ln \left\vert 2\sqrt{a}R + 2ax + b \right\vert + C, \quad a > 0\)
  2. \(\int \frac{dx}{R} = \frac{1}{\sqrt{a}} \sinh^{-1} \frac{2ax + b}{\sqrt{4ac - b^{2}}} + C, \quad a > 0, \; 4ac - b^{2} > 0\)
  3. \(\int \frac{dx}{R} = \frac{1}{\sqrt{a}} \ln \left\vert 2ax + b \right\vert + C, \quad a > 0, \; 4ac - b^{2} = 0\)
  4. \(\int \frac{dx}{R} = - \frac{1}{\sqrt{-a}} \arcsin \frac{2ax + b}{\sqrt{b^{2} - 4ac}} + C, \quad a < 0, \; 4ac - b^{2} < 0\)
  5. \(\int \frac{dx}{R^{3}} = \frac{4ax + 2b}{\left( 4ac - b^{2} \right) \sqrt{R}} + C\)
  6. \(\int \frac{dx}{R^{5}} = \frac{4ax + 2b}{3 \left( 4ac - b^{2} \right) \sqrt{R}} \left( \frac{1}{R} + \frac{8a}{4ac - b^{2}} \right) + C\)
  7. \(\int \frac{dx}{R^{2n + 1}} = \frac{4ax + 2b}{\left( 2n - 1 \right) \left( 4ac - b^{2} \right) R^{\left( 2n - 1 \right)/2}} + \frac{8a \left( n - 1 \right)}{\left( 2n - 1 \right) \left( 4ac - b^{2} \right)} \int \frac{dx}{R^{\left(2n - 1\right)/2}}\)
  8. \(\int \frac{xdx}{R} = \frac{\sqrt{R}}{a} - \frac{b}{2a} \int \frac{dx}{\sqrt{R}}\)
  9. \(\int \frac{xdx}{R^{3}} = \frac{2bx + 4c}{\left( 4ac - b^{2} \right) \sqrt{R}} + C\)
  10. \(\int \frac{xdx}{R^{2n + 1}} = - \frac{1}{\left( 2n - 1 \right) a R^{\left( 2n - 1 \right)/2}} - \frac{b}{2a} \int \frac{dx}{R^{\left( 2n + 1 \right)/2}}\)
  11. \(\int \frac{dx}{xR} = - \frac{1}{\sqrt{c}} \ln \left( \frac{2\sqrt{cR} + bx + 2c}{x} \right) + C, \quad c > 0\)
  12. \(\int \frac{dx}{xR} = \frac{1}{\sqrt{-c}} \sin^{-1} \left( \frac{bx + 2c}{\vert x \vert \sqrt{b^{2} - 4ac}} \right) + C, \quad c < 0\)
  13. \(\int \frac{dx}{xR} = -\frac{1}{\sqrt{c}} \sinh^{-1} \left( \frac{bx + 2c}{\vert x \vert \sqrt{4ac - b^{2}}} \right) + C\)
  14. \(\int \frac{ax^{2} + bx + c}{\sqrt{dx^{2} + ex + c}}\ dx = \sqrt{1 + \left( \frac{2dx + e}{\sqrt{4df - e^{2}}} \right)^{2}} \left( \sqrt{d} \left( 2e \sqrt{4df - e^{2}} - e^{2} \right) \right) + \ln \left\vert \sqrt{1 + \left( \frac{2dx + e}{\sqrt{4df - e^{2}}} \right)^{2}} + \left( \frac{2dx + e}{\sqrt{4df - e^{2}}} \right) \right\vert^{\left( \frac{\sqrt{d}\left( 4df - e^{2} \right)}{2} - c \right)} + \frac{\sqrt{d}\left( 4df - e^{2} \right)}{2}\sqrt{1 + \left( \frac{2dx + e}{\sqrt{4df - e^{2}}} \right)^{2}} \left( \frac{2dx + e}{\sqrt{4df - e^{2}}} \right) + \frac{b}{2d} \left( \sqrt{1 + \left( \frac{2dx + e}{\sqrt{4df - e^{2}}} \right)^{2}} \right) \left( e - \sqrt{4df - e^{2}} \right) + C\)

Integrals involving \( \sqrt{ax + b} \)

  1. \(\int \frac{dx}{x\sqrt{ax + b}} = \frac{-2}{\sqrt{b}} \tanh^{-1} \sqrt{\frac{ax + b}{b}} + C\)
  2. \(\int \frac{\sqrt{ax + b}}{x}\ dx = 2\left( \sqrt{ax + b} - \sqrt{b} \tanh^{-1} \sqrt{\frac{ax + b}{b}} \right) + C\)
  3. \(\int \frac{x^{n}}{\sqrt{ax + b}}\ dx = \frac{2}{a\left( 2n + 1 \right)} \left( x^{n} \sqrt{ax + b} - bn \int \frac{x^{n-1}}{\sqrt{ax + b}} \, dx \right) + C\)
  4. \(\int x^{n}\sqrt{ax + b}\ dx = \frac{2}{2n + 1} \left( x^{n + 1} \sqrt{ax + b} + bx^{n} \sqrt{ax + b} - nb \int x^{n - 1} \sqrt{ax + b} \, dx \right) + C\)

Trigonometric functions

Basic Trigonometric Functions

  1. \(\int \sin(x)\ dx = -\cos(x) + C\)
  2. \(\int \cos(x)\ dx = \sin(x) + C\)
  3. \(\int \tan(x)\ dx = \ln \vert \sec(x) \vert + C\)
  4. \(\int \sin^{2}(x)\ dx = \int \frac{1 - \cos(2x)}{2} = \frac{x}{2} - \frac{\sin(2x)}{4} + C\)
  5. \(\int \cos^{2}(x)\ dx = \int \frac{1 + \cos(2x)}{2} = \frac{x}{2} + \frac{\sin(2x)}{4} + C\)
  6. \(\int \tan^{2}(x)\ dx = \tan(x) - x + C\)

Reciprocal Trigonometric Functions

  1. \(\int \sec(x)\ dx = \ln \left\vert \sec(x) + \tan(x) \right\vert + C = \ln \left\vert \tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\right\vert + C = 2 \operatorname{artanh}\left(\tan\left(\frac{x}{2}\right)\right) + C\)
  2. \(\int \csc(x)\ dx = -\ln \left\vert \csc(x) + \cot(x) \right\vert + C = \ln \left\vert \tan\left(\frac{x}{2}\right) \right\vert + C\)
  3. \(\int \cot(x)\ dx = \ln \vert \sin(x) \vert + C\)
  4. \(\int \sec^{2}(ax)\ dx = \frac{\tan(ax)}{a} + C\)
  5. \(\int \csc^{2}(ax)\ dx = \frac{\cot(ax)}{a} + C\)
  6. \(\int \cot^{2}(ax)\ dx = -x -\frac{\cot(ax)}{a} + C\)
  7. \(\int \sec(x)\tan(x)\ dx = \sec(x) + C\)
  8. \(\int \sec(x)\csc(x)\ dx = \ln\left\vert \tan(x) \right\vert + C\)

Reduction formulae

  1. \(\int \sin^{n}(x)\ dx = - \frac{\sin^{n -1}(x)\cos(x)}{n} + \frac{n - 1}{n}\int\sin^{n - 2}(x)dx + C \quad n > 0\)
  2. \(\int \cos^{n}(x)\ dx = - \frac{\cos^{n -1}(x)\sin(x)}{n} + \frac{n - 1}{n}\int\cos^{n - 2}(x)dx + C \quad n > 0\)
  3. \( \int\tan^{n}(x)\ dx = \frac{\tan^{n - 1}(x)}{n - 1} - \int\tan^{n - 2}(x)dx + C \quad n \neq 1\)
  4. \(\int \sec^{n}(x)\ dx = \frac{\sec^{n - 1}(x)\sin(x)}{n - 1} + \frac{n -2}{n - 1} \int \sec^{n - 2}(x)dx + C \quad n \neq 1\)
  5. \(\int \csc^{n}(x)\ dx = - \frac{\csc^{n - 1}(x)\cos(x)}{n - 1} + \frac{n -2}{n - 1} \int \csc^{n - 2}(x)dx + C \quad n \neq 1\)
  6. \(\int \cot^{n}(x)\ dx = - \frac{\cot^{n - 1}(x)}{n - 1} + \int \cot^{n - 2}(x)dx + C \quad n \neq 1\)
  7. \(\int a^{2}x^{n}\sin(ax)\ dx = nx^{n - 1}\sin(ax) - ax^{n}\cos(ax) - n(n - 1) \int x^{n - 2} \sin(ax)dx\)
  8. \(\int a^{2}x^{n}\cos(ax)\ dx = ax^{n}\sin(ax) + nx^{n - 1}\cos(ax) - n(n - 1) \int x^{n - 2} \cos(ax)dx\)

Explicit forms

  1. \(\int \sin^{n}(x)\ dx = - \cos(x)_{2}F_{1}\left( \frac{1}{2},\frac{1 - n}{2};\frac{3}{2};\cos^{2}(x) \right) + C\)
  2. \(\int \cos^{n}(x)\ dx = - \frac{1}{n + 1} \operatorname{sgn}(\sin(x))\cos^{n + 1}(x)_{2}F_{1}\left( \frac{1}{2},\frac{n + 1}{2};\frac{n + 3}{2};\cos^{2}(x) \right) + C \quad n \neq -1\)
  3. \(\int \tan^{n}(x)\ dx = \frac{1}{n + 1} \tan^{n + 1}(x)_{2}F_{1}\left( 1,\frac{n + 1}{2};\frac{n + 3}{2};-\tan^{2}(x) \right) + C \quad n \neq -1\)
  4. \(\int \csc^{n}(x)\ dx = -\cos(x)_{2}F_{1} \left( \frac{1}{2},\frac{n + 1}{2};\frac{3}{2};\cos^{2}(x)\right) + C\)
  5. \(\int \sec^{n}(x)\ dx = \sin(x)_{2}F_{1} \left( \frac{1}{2},\frac{n + 1}{2};\frac{3}{2};\sin^{2}(x)\right) + C\)
  6. \(\int \cot^{n}(x)\ dx = -\frac{1}{n + 1} \cot^{n + 1}(x)_{2}F_{1}\left( 1,\frac{n + 1}{2};\frac{n + 3}{2};-\cot^{2}(x) \right) + C \quad n \neq -1\)

Where \(_{2}F_{1}\) is the hypergeometric function and \( \operatorname{sgn} \) is the sign function.

Irrational functions

  1. \(\int \frac{dx}{\sqrt{1 - x^{2}}} = \operatorname{arcsin}(x) + C\)
  2. \(\int \frac{dx}{\sqrt{a^{2} - x^{2}}} = \operatorname{arcsin}\left(\frac{x}{a}\right) + C \quad a \neq 0\)
  3. \(\int \frac{dx}{1 - x^{2}} = \operatorname{arctan}(x) + C\)
  4. \(\int \frac{dx}{a^{2} + x^{2}} = \frac{\operatorname{arctan}(x)}{a} + C \quad a \neq 0\)

Exponential and logarithmic functions

  1. \(\int e^{x}\ dx = e^{x} + C\)
  2. \(\int e^{ax}\ dx = \frac{e^{ax}}{a} + C\)
  3. \(\int a^{x}\ dx = \frac{a^{x}}{\ln(a)} + C\)
  4. \(\int \ln x\ dx = \log_{e}x = x \ln x - x + C\)
  5. \(\int e^{x}\sin(x)\ dx = \frac{e^{x}}{2}\left( \sin(x) - \cos(x) \right) + C\)
  6. \(\int e^{x}\cos(x)\ dx = \frac{e^{x}}{2}\left( \sin(x) + \cos(x) \right) + C\)

Reduction formulae

  1. \(\int x^{n}e^{ax}\ dx = \frac{1}{a} x^{n}e^{ax} - \frac{n}{a}\int x^{n-1}e^{ax}\ dx\)

Inverse trigonometric functions

  1. \(\int \operatorname{arcsin}(x)\ dx = x \operatorname{arcsin}(x) + \sqrt{1 - x^{2}} + C\)
  2. \(\int \operatorname{arccos}(x)\ dx = x \operatorname{arccos}(x) - \sqrt{1 - x^{2}} + C\)
  3. \(\int \operatorname{arctan}(x)\ dx = x \operatorname{arctan}(x) - \frac{1}{2} \ln\vert 1 + x^{2}\vert + C\)
  4. \(\int \operatorname{arccsc}(x)\ dx = x \operatorname{arccsc}(x) + \ln\left\vert x + x \sqrt{1 - \frac{1}{x^2}} \right\vert + C\)
  5. \(\int \operatorname{arcsec}(x)\ dx = x \operatorname{arcsec}(x) - \ln\left\vert x + x \sqrt{1 - \frac{1}{x^2}} \right\vert + C\)
  6. \(\int \operatorname{arccot}(x)\ dx = x \operatorname{arccot}(x) + \sqrt{1 - x^{2}} + C\)

Hyperbolic functions

  1. \(\int \sinh(x)\ dx = -i \int \sin(ix)\ dx = \cos(ix) + C = \cosh(x) + C\)
  2. \(\int \cosh(x)\ dx = \int \cos(ix)\ dx = -i\sin(ix) + C = \sinh(x) + C\)
  3. \(\int \tanh(x)\ dx = -i \int \tan(ix)\ dx = \log \vert \cos(ix)\vert + C = \log \vert \cosh(x)\vert + C\)

Reciprocals

  1. \(\int \operatorname{csch}(x) = i \int \csc(ix) = \log \left\vert -i \tan\left( \frac{ix}{2} \right) \right\vert + C = \log\left\vert \tanh\left( \frac{x}{2} \right) \right\vert + C\)
  2. \(\int \operatorname{sech}(x) = \int \sec(ix) = 2 \operatorname{artanh}\left( -i \tan\left( \frac{x}{2}i \right) \right)+ C = 2 \operatorname{arctan}\left( \tanh \left( \frac{x}{2} \right) \right) + C\)
  3. \(\int \operatorname{coth}(x) = i \int \cot(ix) = \log\vert -i \sin(ix) \vert + C = \log\vert \sinh(x) \vert + C\)

Inverses

  1. \(\int \operatorname{arsinh}(x)\ dx = x\operatorname{arsinh}(x) - \sqrt{x^{2} + 1} + C\)
  2. \(\int \operatorname{arcosh}(x)\ dx = x\operatorname{arcosh}(x) - \sqrt{x^{2} - 1} + C\)
  3. \(\int \operatorname{artanh}(x)\ dx = x\operatorname{artanh}(x) + \frac{1}{2} \ln(1 - x^{2}) + C\)
  4. \(\int \operatorname{arcsch}(x)\ dx = x\operatorname{arcsch}(x) + \vert \operatorname{arsinh}(x) \vert + C\)
  5. \(\int \operatorname{arsech}(x)\ dx = x\operatorname{arsech}(x) + \operatorname{arcsin}(x) + C\)
  6. \(\int \operatorname{arcoth}(x)\ dx = x\operatorname{arcoth}(x) + \frac{1}{2} \ln(x^{2} - 1) + C\)

Misc

  1. \(\int \vert f(x) \vert\ dx = \operatorname{sgn}\left( f(x) \right) \int f(x)dx\)

Definite integrals

  • \(\int_{[0,1]^{n}}\frac{\Pi^{n}_{i = 1}dx_{i}}{1 - \Pi^{n}_{i = 1}x_{i}} = \zeta(n)\text{ for all integers } n > 1\), where \(\zeta\) is the Riemann zeta function
  • \(\int_{-\infty}^{\infty} e^{-x^{2}}dx = \sqrt{\pi}\)
  • \(\int_{0}^{1}t^{u - 1}(1 - t)^{v - 1}dt = \beta(u,v) = \frac{\Gamma(u)\Gamma(v)}{\Gamma(u + v)}\), where \(\Gamma\) is the gamma function.
  • \(\int_{0}^{\infty}t^{s-1}e^{-t}dt = \Gamma(s)\)
  • \(\int_{0}^{2 \pi} e^{u cos\theta}d\theta = 2\pi I_{0}(u)\), where \( I_{0} \) is the modified Bessel function of the first kind.
  • \(\int_{0}^{2 \infty}\frac{\sin(x)}{x} = \frac{\pi}{2}\)

See also

Differential Calculus