Properties of Definite Integration

Properties of Definite Integration:

1. Linearity Property:

  • Statement: If f(x) and g(x) are integrable functions and c is a constant, then ab[cf(x)+g(x)]dx=cabf(x)dx+abg(x)dx
  • This property allows you to split an integral over the sum or difference of functions into separate integrals.

2. Additivity Property:

  • Statement: If f(x) is integrable on [a,b] and c is a constant, then abf(x)dx=acf(x)dx+cbf(x)dx
  • This property allows the integral over an interval to be split into two integrals over sub-intervals.

3. Constant Multiple Property:

  • Statement: If f(x) is integrable on [a,b] and c is a constant, then abcf(x)dx=cabf(x)dx
  • This property allows a constant factor to be pulled out of the integral.

4. Symmetry Property:

  • Statement: If f(x) is integrable on [a,a], then aaf(x)dx=0
  • This property is helpful when dealing with functions that exhibit symmetry around the y-axis.

5. Reversal of Limits Property:

  • Statement: If f(x) is integrable on [a,b], then abf(x)dx=baf(x)dx
  • This property allows you to switch the limits of integration while introducing a negative sign.

6. Integral of a Constant:

  • Statement: For any constant c, abcdx=c(ba)
  • The integral of a constant function over an interval is the constant multiplied by the length of the interval.

7. Integration by Parts for Definite Integrals:

  • Statement: For two functions u(x) and v(x) differentiable on [a,b], abu(x)dv(x)=[u(x)v(x)]ababv(x)du(x)
  • Integration by parts can be applied to definite integrals, but the boundary terms need to be accounted for in the evaluation.

8. Change of Variable Property (Substitution Rule):

  • Statement: If u=g(x) is a differentiable function with a continuous derivative on [a,b], abf(g(x))g(x)dx=g(a)g(b)f(u)du
  • If u=g(x) is a differentiable function on [a,b]and f(g(x))g(x) is integrable on [a,b], then g(a)g(b)f(u)du=abf(g(x))g(x)dx.
  • This property helps in simplifying integrals by changing the variable of integration.

9. Inequality Property:

If f(x)g(x) for all x in [a,b], then abf(x)dxabg(x)dx.

10. Absolute Value Property:

abf(x)dxabf(x)dx.

11. Additivity over Subintervals Property:

If a function f(x) is integrable over subintervals [a,b] and [b,c], then acf(x)dx=abf(x)dx+bcf(x)dx.

12. Periodicity Property:

For a function f(x) with period T, aa+Tf(x)dx=0Tf(x)dx for any a.