Definite Integral | Mathematics (original) (raw)

Last Updated : 11 Jul, 2025

Definite integrals are the extension after indefinite integrals, definite integrals have limits [a, b]. It gives the area of a curve bounded between given limits. \int_{a}^{b}F(x)dx , It denotes the area of curve F(x) bounded between a and b, where a is the lower limit and b is the upper limit.

Note: If f is a continuous function defined on the closed interval [a, b] and F be an anti derivative of f.

Then \int_{a}^{b}f(x)dx= \left [ F(x) \right ]_{a}^{b}\right = F(b)-F(a) Here, the function f needs to be well defined and continuous in [a, b].

Example: Find, \int_{1}^{4}x^{2}dx ? Solution: Since, \int x^{2}=\frac{x^{3}}{3} \newline \newline \textup{Then F(x)} =\frac{x^{3}}{3} \newline \newline [F(x)]_{1}^{4}= F(4)-F(1) \newline \newline =[\frac{4^{3}}{3} - \frac{1^{3}}{3}]=\frac{65}{3}

  1. \int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt
  2. \int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx
  3. \int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx
  4. \int_{a}^{b}f(x)=\int_{a}^{b}f(a+b-x)dx
  5. \int_{0}^{b}f(x)=\int_{0}^{b}f(b-x)dx
  6. \int_{0}^{2a}f(x)dx=\int_{0}^{a}f(x)dx+\int_{0}^{a}f(2a-x)dx
  7. \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx, \textup{if f(x) is even function i.e f(x)=f(-x)}
  8. \int_{-a}^{a}f(x)dx=0, \textup{if f(x) is odd function}
  9. \bold{\int\limits_{0}^{2a}f(x)dx = \begin{cases} 2\int\limits_{0}^{a}f(x)dx & , if f(2a - x) = f(x) \\ 0 & , if f(2a - x) = -f(x)\end{cases}}

These properties can be used directly to find the value of a particular definite integral and also interchange to other forms if required.