Question

If f is continuous for all x, which of the following integrals necessarily have the same value?

Answer 1

Answer:

B

Step-by-step explanation:

Given the three integrals, we want to determine which integrals necessarily have the same value.

We can let the first integral be itself.

For the second integral, we can perform a u-substitution. Let u = x + a. Then:

$\displaystyle du = dx$

Changing our limits of integration:

$u_1=(0)+a=a \text{ and } u_2 = (b+a)+a = b+2a$

Thus, the second integral becomes:

$\displaystyle \int_{0}^{b+a}f(x+a)\, dx = \int_a^{b+2a} f(u)\, du$

For the third integral, we can also perform a u-substitution. Let u = x + c. Then:

$\displaystyle du = dx$

And changing our limits of integration:

$\displaystyle u_1=(a-c)+c=a \text{ and } u_2=(b-c)+c=b$

Thus, our third integral becomes:

$\displaystyle \int_{a-c}^{b-c}f(x+c)\, dx = \int_{a}^{b} f(u)\, du$

Since the only difference between f(x) and f(u) is the variable and both the first and third integral have the same limits of integration, our answer is B.