Integral calculus - Indefinite integral

  1. \(\int f(x)dx=F(x)+C \ \ if \ \ F' (x) = f(x).\) 
  2. \((\int f(x)dx )'=f(x)\)
  3. \(\int kf (x) dx = k \int f(x) dx\)
  4. \(\int [f(x)+g(x)]dx= \int f(x)dx+ \int g(x)dx\)
  5. \(\int [f(x)-g(x)]dx= \int f(x)dx- \int g(x)dx\)
  6. \(\int [f(ax)dx= {1 \over a} F(ax)+C\)
  7. \(\int f(ax+b)dx= {1 \over a}F (ax+b ) +C\)
  8. \(\int f(x)f'(x)dx={1 \over 2}f^2(x)+c\)
  9. \(\int {\text {f'(x)} \over f(x)}dx |= In | f(x) |+C\)
  10. Method of substitution
    \(\int f(x)dx= \int f(u(t))u'(t)dt\:\: \:if \:=x=u(t).\)
  11. Integration by Parts
    \(\int udv=uv- \int vdu,\)
    Where u(x), v(x) are differentiable functions.