What is the Difference Between Definite and Indefinite Integrals?
Smaridasa Mathematics – After understanding the definition of the integral, and before we go further into integrals, it is good to know first What is the Difference Between Definite and Indefinite Integrals?. The fundamental difference between definite integrals and indefinite integrals lies in the limits of integration. In this article, we will briefly discuss the differences between the two and will explore them in more detail in other articles.
Integral Notation
Definite integral: \( \int_a^b f(x) \, dx \)
Indefinite integral: \( \int f(x) \, dx \)
Differences Between Definite and Indefinite Integrals
The differences between the two integrals are as follows:
| Aspect | Definite Integral | Indefinite Integral |
|---|---|---|
| Limits of Integration | Has lower limit \( a \) and upper limit \( b \) | No limits of integration |
| Result | Usually a number (depending on the limits) | A function (family of functions) |
| Notation | \( \displaystyle \int_a^b f(x) \, dx \) | \( \displaystyle \int f(x) \, dx \) |
| Constant of Integration | No \( + c \) in the result | Includes \( + c \) (constant of integration) |
Mathematically:
\[ \int_a^b f(x) \, dx = \big[ F(x) \big]_a^b = F(b) - F(a) \]
\[ \int f(x) \, dx = F(x) + c \]
Note:
For both definite and indefinite integrals, we must first find the integral of the function. Therefore, we should master indefinite integrals first, after which it will be easier to work on definite integrals by directly substituting the limits.
Example Problems:
1) Determine the result of the following integrals:
a) \( \displaystyle \int 2x \, dx \) (indefinite)
b) \( \displaystyle \int_{2}^{7} 2x \, dx \) (definite)
Solution (a) – Indefinite Integral:
We know from the definition of the integral that it is the antiderivative or the inverse of the derivative.
\[ \int 2x \, dx = x^2 + c \]
Because the derivative of \( x^2 + c \) is \( 2x \).
Thus, the result of \( \int 2x \, dx = x^2 + c \).
Solution (b) – Definite Integral:
First, we find the indefinite integral \( \int 2x \, dx \), then substitute the upper and lower limits.
From part (a) above, we have \( \int 2x \, dx = x^2 + c \).
\[ \int_{2}^{7} 2x \, dx = \big[ x^2 \big]_{2}^{7} \]
\[ = F(7) - F(2) \]
\[ = 7^2 - 2^2 \]
\[ = 49 - 4 \]
\[ = 45 \]
Therefore, \( \displaystyle \int_{2}^{7} 2x \, dx = 45 \).