Question

Which of the following is NOT a definite integral?
A. $\int\limits_{ - 5}^5 {f(x)} dx$
B. $\int {g(y)} dy$
C. $\int\limits_0^5 {} du$
D. \[\int\limits_5^6 0 dx\]

Answer 1

Hint: In this type of question, we should have a good knowledge of definite integrals. The approach we will use for this question will check every option and try to match it up with the basic definition of definite integral.

Complete step by step solution:
Definite integrals are those integrals that have an upper and lower limit. Definite integral has two different values for the upper limit and lower limit. The final value of a definite integral is the value of integral to the upper limit minus the value of the definite integral for the lower limit.
It is represented as $\int\limits_a^b {f(x)} = f(b) - f(a)$ in a continuous limit $(a,b)$ where $a \leqslant x \leqslant b$
Option A has upper limit ranging from $-5$ to $ 5$
Option C also has the defined and continuous limit i.e., from $0$ to $5$.
In option D, although we are getting 0, still it has continuous limits. That’s why it is a definite integral.
Looking toward the options we can say that which option doesn’t have an upper limit and lower limit.
Hence option B is correct.

Note: Properties of Definite Integrals:
Property 1: Definite integrals between the same limits of the same function with different variables are equal.
$\int\limits_a^b {f(p)} dp = \int\limits_a^b {f(q)} dq$
Property 2: The value of a definite integral is equal to its negative when the upper and lower limits of the integral function are interchanged.
$\int\limits_a^b {f(p)} dp = - \int\limits_b^a {f(q)} dq$
Property 3: The definite integral of a function is zero when the upper and lower limits are the same.
$\int\limits_a^a {f(p)} dp = 0$
Property 4: A definite integral can be written as the sum of two definite integrals. However, the following conditions must be considered.
a. The lower limit of the first addend should be equal to the lower limit of the original definite integral.
b. The upper limit of the second addend should be equal to the upper limit of the original definite integral.
c. The upper limit of the first addend should be equal to the lower limit of the second addend integral.