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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 $$\Delta {x_i} = {x_i} – {x_{i – 1}},$$ $${x_{i – 1}} \le {\xi _i} \le {x_i}.$$
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
$${\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =$$ $$\large\frac{{b – a}}{{2n}}\normalsize\Big[ {f\left( {{x_0}} \right) + f\left( {{x_n}} \right) }$$ $$+\;{ 2\sum\limits_{i = 1}^{n – 1} {f\left( {{x_i}} \right)} } \Big]$$
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 $${x_i} = a + {\large\frac{{b – a}}{n}\normalsize} i,$$ $$i = 0,1,2, \ldots ,n.$$
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 $$F^{\,\prime}\left( x \right) = f\left( x \right).$$
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 $$F^{\,\prime}\left( x \right) = f\left( x \right)$$, $$G^{\,\prime}\left( x \right) = g\left( x \right).$$