Definite Integration-Integration Formulas

Definite Integration

Integrands: f, g, u, v

Antiderivatives: F, G

Independent variables: x, t

Limits of integration: a, b, c, d

Subintervals of integration: Δxi

Arbitrary point of a subinterval: ξi

Natural numbers: n, i

Area of a curvilinear trapezoid: S

2. The definite integral of 1 is equal to the length of the interval of integration:

$\underset{a}{\overset{b}{\int }}1dx=b–a$

3. A constant factor can be moved across the integral sign:

$\underset{a}{\overset{b}{\int }}kf\left(x\right)dx=k\underset{a}{\overset{b}{\int }}f\left(x\right)dx$

4. The definite integral of the sum of two functions is equal to the sum of the integrals of these functions:

$\underset{a}{\overset{b}{\int }}\left[f\left(x\right)+g\left(x\right)\right]dx=\underset{a}{\overset{b}{\int }}f\left(x\right)dx+\underset{a}{\overset{b}{\int }}g\left(x\right)dx$

5. The definite integral of the difference of two functions is equal to the difference of the integrals of these functions:

$\underset{a}{\overset{b}{\int }}\left[f\left(x\right)–g\left(x\right)\right]dx=\underset{a}{\overset{b}{\int }}f\left(x\right)dx–\underset{a}{\overset{b}{\int }}g\left(x\right)dx$

6. If the upper and lower limits of a definite integral are the same, the integral is zero:

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=0$

7. Reversing the limits of integration changes the sign of the definite integral:

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=–\underset{b}{\overset{a}{\int }}f\left(x\right)dx$

8. Suppose that a point c belongs to the interval [a,b]. Then the definite integral of a function f(x) over the interval [a,b] is equal to the sum of the integrals over the intervals [a,c] and [c,b]:

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{c}{\int }}f\left(x\right)dx+\underset{c}{\overset{b}{\int }}f\left(x\right)dx$

12. Substitution rule for definite integrals

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{c}{\overset{d}{\int }}f\left(g\left(t\right)\right)g‘\left(t\right)dt$

13. Integration by parts for definite integrals

$\underset{a}{\overset{b}{\int }}udv=\left(uv\right){|}_{a}^{b}–\underset{a}{\overset{b}{\int }}vdu$

14. Trapezoidal approximation of a definite integral

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\frac{b–a}{2n}\left[f\left({x}_{0}\right)+f\left({x}_{n}\right)+2\sum _{i=1}^{n–1}f\left({x}_{i}\right)\right]$

15. Approximation of a definite integral using Simpson’s rule

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\frac{b–a}{3n}\left[f\left({x}_{0}\right)+4f\left({x}_{1}\right)+2f\left({x}_{2}\right)+4f\left({x}_{3}\right)+\cdots +4f\left({x}_{n–1}\right)+f\left({x}_{n}\right)\right],$

16. Area under a curve

$S=\underset{a}{\overset{b}{\int }}f\left(x\right)dx=F\left(b\right)–F\left(a\right),$

17. Area between two curves

$S=\underset{a}{\overset{b}{\int }}\left[f\left(x\right)–g\left(x\right)\right]dx=F\left(b\right)–G\left(b\right)–F\left(a\right)+G\left(a\right)$