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.
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
2 answers
I and III only