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Q5)

for any function f(x). True False True False True False True False Let f be a differentiable function with the following values: After substitution, which of the following integrals is equivalent to Evaluate the integral. Give an exact answer. After substitution, which of the following integrals is equivalent to Evaluate the integral. Give an exact answer.

Explanation / Answer

1. False

2. False

3. False: the upper limit is wrong

4. False

5. (a) substitute u = f(x)

lower limit = f(0) = 5

upper limit = f(1) = 8

So, (C) is the answer = int( sin(u) du ) from 5 to 8

exact answer = cos(8) - cos(5) = -0.429

(b) substitute u = cosx

du = -sinx dx

lower limit = cos(0) = 1

upper limit = cos(pi/2) = 0

so, ans is: int( - f'(u) du ) from 1 to 0

that is: int( f'(u) du ) from 0 to 1 : option (D)

exact answer: f(1) - f(0) = 8 - 5 = 3

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