Question

If f(x) is continuous for all x, which of the following integrals necessarily have the same value?
b
I. ∫ f(x) dx
a

b
II. ∫ |f(x)| dx
a

b-c
III. ∫ f(x+c) dx
a-c

b
IV. ∫ (f(x)+c) dx
a


I and II only

I and III only

I, II and IV only

II, III, and IV only

Answer 1

not II. Consider f(x) = ∫[0,1] x^2-1
not IV. If F(x) = ∫f(x) dx then
∫(f(x)+c) dx = ∫f(x) dx + cx

Now, for III, let u = x+c
Then you have
∫[a,b] f(u) du
Makes sense, since f(x+c) is shifted to the left by c, and so is the interval of integration.

Answer 2

I and III only