If g(x) is continuous for all x, which of the following integrals necessarily hv the same value?
Solved
I think that 1 and 2 won't have the same value, but I am not completely sure. I have no idea when comparing 1 and 2 to 3 and 4.
If g(x) is continuous for all x, which of the following integrals necessarily hv the same value?
Solved
I think that 1 and 2 won't have the same value, but I am not completely sure. I have no idea when comparing 1 and 2 to 3 and 4.
For 3. substitute u = x + c. Then:
$\displaystyle \int_{a-c}^{b-c} f(x + c) \, dx = \int_{a}^{b} f(u) \, du$ which is actually equivalent to $\displaystyle \int_{a}^{b} f(x) \, dx$ since a definite integral does not depend on the (dummy) variable used.
The correct answer is therefore option B.
To show that 1. and 2. are not equal, a simple counter-example is sufficient. Consider f(x) = -x, a = 0 and b = 1.
For 4. note that
$\displaystyle \int_{a}^{b} f(x) + c \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b}c \, dx = \int_{a}^{b} f(x)\, dx + c(b - a) \neq \int_{a}^{b} f(x)\, dx$ unless a = b or c = 0.