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APEX Calculus

Section B.2 Integration Formulas

List B.2.1. Basic Rules
  1. \(\displaystyle \int c\cdot f(x)\,dx=c\int f(x)\,dx\)
  2. \(\displaystyle \int \bigl(f(x)\pm g(x)\bigr)\,dx = \int f(x)\, dx \pm \int g(x)\, dx\)
  3. \(\displaystyle \int 0\,dx = C\)
  4. \(\displaystyle \int 1\,dx=x+C\)
List B.2.2. Integrals of Elementary (non-Trig) Functions
  1. \(\displaystyle \int e^x\,dx=e^x+C\)
  2. \(\displaystyle \int \ln x\,dx=x\ln x -x +C\)
  3. \(\displaystyle \int a^x\,dx=\frac{1}{\ln a}\cdot a^x+C\)
  4. \(\displaystyle \int \frac{1}{x}\,dx =\ln \abs{x} + C\)
  5. \(\displaystyle \int x^n\,dx=\frac{1}{n+1}x^{n+1}+C, n\neq -1\)
List B.2.3. Integrals Involving Trigonometric Functions
  1. \(\displaystyle \int \cos x\,dx=\sin x+C\)
  2. \(\displaystyle \int \sin x\,dx=-\cos x+C\)
  3. \(\displaystyle \int \tan x\,dx=-\ln \abs{\cos x}+C\)
  4. \(\displaystyle \int \sec x\,dx=\ln \abs{\sec x+\tan x}+C\)
  5. \(\displaystyle \int \csc x\,dx=-\ln \abs{\csc x+\cot x}+C\)
  6. \(\displaystyle \int \cot x\,dx=\ln \abs{\sin x}+C\)
  7. \(\displaystyle \int \sec^2 x\,dx=\tan x+C\)
  8. \(\displaystyle \int \csc^2x\,dx=-\cot x+C\)
  9. \(\displaystyle \int \sec x\tan x\,dx=\sec x+C\)
  10. \(\displaystyle \int \csc x\cot x\,dx=-\csc x+C\)
  11. \(\displaystyle \int \cos^2x\,dx=\frac12x+\frac14\sin\big(2x\big)+C\)
  12. \(\displaystyle \int \sin^2x\,dx=\frac12x-\frac14\sin\big(2x\big)+C\)
  13. \(\displaystyle \int \frac{1}{x^2+a^2}\,dx = \frac1a\tan^{-1}\left(\frac xa\right)+C\)
  14. \(\displaystyle \int \frac{1}{\sqrt{a^2-x^2}} = \sin^{-1}\left(\frac xa\right)+C\)
  15. \(\displaystyle \int \frac{1}{x\sqrt{x^2-a^2}} = \frac1a\sec^{-1}\left(\frac{\abs{x}}{a}\right)+C\)
List B.2.4. Integrals Involving Hyperbolic Functions
  1. \(\displaystyle \int \cosh x\,dx=\sinh x+C\)
  2. \(\displaystyle \int \sinh x\,dx=\cosh x+C\)
  3. \(\displaystyle \int \tanh x\,dx=\ln(\cosh x)+C\)
  4. \(\displaystyle \int \coth x\,dx=\ln \abs{\sinh x}+C\)
  5. \(\displaystyle \int \frac{1}{\sqrt{x^2-a^2}}\, dx =\ln\abs{x+\sqrt{x^2-a^2}}+C\)
  6. \(\displaystyle \int \frac{1}{\sqrt{x^2+a^2}}\, dx=\ln\abs{x+\sqrt{x^2+a^2}}+C\)
  7. \(\displaystyle \int \frac{1}{a^2-x^2}\, dx =\frac{1}{2a}\ln\abs{\frac{a+x}{a-x}}+C\)
  8. \(\displaystyle \int \frac{1}{x\sqrt{a^2-x^2}}\, dx = \frac{1}{a}\ln\left(\frac{x}{a+\sqrt{a^2-x^2}}\right)+C\)
  9. \(\displaystyle \int \frac{1}{x\sqrt{x^2+a^2}}\, = \frac{1}{a}\ln\abs{\frac{x}{a+\sqrt{x^2+a^2}}}+C\)
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