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Question 1:
True or False. The second fundamental theorem of calculus states that if
F(x) = ò a x f(t) dt
then F '(x) = f(x).
Answer :
True.
Question 2:
True or False. If
F(x) = ò -2 3x sin(t) dt
then the second fundamental theorem of calculus can be used to evaluate F '(x) as follows
F '(x) = sin (3x)
Answer :
False.
Note that the upper limit in the integral above is 3x and not x, hence the integral above has the form
F(x) = ò -2 u(x) f(t) dt
Using the chain rule, we can write
F '(x) = dF / du * du / dx = 3 sin (3x)
Question 3:
True or False. Using the first fundamental of calculus
òa b f(x) dx = F(b) - F(a)
we can evaluate the following integral as follows
ò-1 1 (1 / x2) dx = -2
Answer :
False. The interval of integration [-1 , 1] contains 0 at which function 1 / x2 is discontinuous and the above theorem cannot be applied.
More references on calculus
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