Formulas and Tables

Calculus

Properties of Definite Integrals

Integrands: \(f\), \(g\), \(u\), \(v\)
Antiderivatives: \(F\), \(G\)
Independent variables: \(x\), \(t\)
Limits of integration: \(a\), \(b\), \(c\), \(d\)

Subintervals of integration: \(\Delta {x_i}\)
Arbitrary point of a subinterval: \({\xi_i}\)
Natural numbers: \(n\), \(i\)
Area of a curvilinear trapezoid: \(S\)

  1. Let a real function \(f\left( x \right)\) be defined and bounded on the interval \(\left[ {a,b} \right]\). Let us divide this interval into \(n\) subintervals. In each interval, we choose an arbitrary point \({\xi_i}\) and form the integral sum \(\sum\limits_{i = 1}^n {f\left( {{\xi _i}} \right)\Delta {x_i}}\) where \(\Delta {x_i}\) is the length of the \(i\)th interval. The definite integral of the function \(f\left( x \right)\) over the interval \(\left[ {a,b} \right]\) is defined as the limit of the integral sum (Riemann sums) as the maximum length of the subintervals approaches zero.
    \(\require{AMSmath.js}
    {\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \(\lim\limits_{\substack{
    n \to \infty\\
    \text{max}\,\Delta {x_i} \to 0}} \sum\limits_{i = 1}^n {f\left( {{\xi _i}} \right)\Delta {x_i}} ,\) where \(\Delta {x_i} = {x_i} – {x_{i – 1}},\) \({x_{i – 1}} \le {\xi _i} \le {x_i}.\)
The definite integral of a function is defined as the limit of the integral sum
  1. The definite integral of \(1\) is equal to the length of the interval of integration:
    \({\large\int\limits_a^b\normalsize} {1\,dx} = b – a\)
  2. A constant factor can be moved across the integral sign:
    \({\large\int\limits_a^b\normalsize} {kf\left( x \right)dx} =\) \( k{\large\int\limits_a^b\normalsize} {f\left( x \right)dx} \)
  3. The definite integral of the sum of two functions is equal to the sum of the integrals of these functions:
    \({\large\int\limits_a^b\normalsize} {\left[ {f\left( x \right) + g\left( x \right)} \right]dx} =\) \( {\large\int\limits_a^b\normalsize} {f\left( x \right)dx} + {\large\int\limits_a^b\normalsize} {g\left( x \right)dx} \)
  4. The definite integral of the difference of two functions is equal to the difference of the integrals of these functions:
    \({\large\int\limits_a^b\normalsize} {\left[ {f\left( x \right) – g\left( x \right)} \right]dx} =\) \( {\large\int\limits_a^b\normalsize} {f\left( x \right)dx} – {\large\int\limits_a^b\normalsize} {g\left( x \right)dx} \)
  5. If the upper and lower limits of a definite integral are the same, the integral is zero:
    \({\large\int\limits_a^a\normalsize} {f\left( x \right)dx} = 0\)
  6. Reversing the limits of integration changes the sign of the definite integral:
    \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \( -{\large\int\limits_b^a\normalsize} {f\left( x \right)dx}\)
  7. Suppose that a point \(c\) belongs to the interval \(\left[ {a,b} \right]\). Then the definite integral of a function \(f\left( x \right)\) over the interval \(\left[ {a,b} \right]\) is equal to the sum of the integrals over the intervals \(\left[ {a,c} \right]\) and \(\left[ {c,b} \right]:\)
    \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \( {\large\int\limits_a^c\normalsize} {f\left( x \right)dx} \) \(+\; {\large\int\limits_c^b\normalsize} {f\left( x \right)dx}\)
  8. The definite integral of a non-negative function is always greater than or equal to zero:
    \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} \ge 0\) if \(f\left( x \right) \ge 0 \text{ in }\left[ {a,b} \right].\)
  9. The definite integral of a non-positive function is always less than or equal to zero:
    \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} \le 0\) if \(f\left( x \right) \le 0 \text{ in } \left[ {a,b} \right].\)
  10. Fundamental theorem of calculus
    \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \( {\left. {F\left( x \right)} \right|_a^b} =\) \( F\left( b \right) – F\left( a \right),\) if \(F’\left( x \right) = f\left( x \right).\)
  11. Substitution rule for definite integrals
    If \(x = g\left( t \right)\), then \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \({\large\int\limits_c^d\normalsize} {f\left( {g\left( t \right)} \right)g’\left( t \right)dt}, \) where \(c = {g^{ – 1}}\left( a \right),\) \(d = {g^{ – 1}}\left( b \right).\)
  12. Integration by parts for definite integrals
    \({\large\int\limits_a^b\normalsize} {udv} =\) \(\left.{\left( {uv} \right)}\right|_a^b – {\large\int\limits_a^b\normalsize} {vdu} \)
  13. Trapezoidal approximation of a definite integral
    \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \(\large\frac{{b – a}}{{2n}}\normalsize\Big[ {f\left( {{x_0}} \right) + f\left( {{x_n}} \right) }\) \(+\;{ 2\sum\limits_{i = 1}^{n – 1} {f\left( {{x_i}} \right)} } \Big]\)
Trapezoidal approximation of a definite integral
  1. Approximation of a definite integral using Simpson’s rule
    \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \({\large\frac{{b – a}}{{3n}}\normalsize}\Big[ {f\left( {{x_0}} \right) + 4f\left( {{x_1}} \right) }\) \(+\;{2f\left( {{x_2}} \right)}\) \(+\;{ 4f\left( {{x_3}} \right) + 2f\left( {{x_4}} \right) + \ldots}\) \(+\;{4f\left( {{x_{n – 1}}} \right) + f\left( {{x_n}} \right)} \Big],\)
    where \({x_i} = a + {\large\frac{{b – a}}{n}\normalsize} i,\) \(i = 0,1,2, \ldots ,n.\)
Approximation of a definite integral using Simpson’s rule
  1. Area under a curve
    \(S = {\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \(F\left( b \right) – F\left( a \right),\) where \(F^{\,\prime}\left( x \right) = f\left( x \right).\)
Area under a curve
  1. Area between two curves
    \(S = {\large\int\limits_a^b\normalsize} {\left[ {f\left( x \right) – g\left( x \right)} \right]dx} =\) \(F\left( b \right) – G\left( b \right) \) \(-\;F\left( a \right) + G\left( a \right),\) where \(F^{\,\prime}\left( x \right) = f\left( x \right)\), \(G^{\,\prime}\left( x \right) = g\left( x \right).\)
Area between two curves