Let f and g be continuous functions such that \(\int_{0}^{2}\displaystyle f(x)\,dx = 6\), \(\int_{0}^{5}\displaystyle f(x)\,dx = 10\) and \(\int_{2}^{5}\displaystyle g(x)\,dx = 3\). Which of the following statements are correct?
(A) \(\int_{2}^{5}\displaystyle f(x)\,dx = 4\)
(B) \(\int_{5}^{2}\displaystyle f(x)\,dx = 4\)
(C) \(\int_{2}^{5}\displaystyle\left[2f(x)-g(x)\right]dx = 5\)
(D) \(\int_{0}^{2}\displaystyle 3f(x)\,dx = 2\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
By splitting the interval at 2,
\( \displaystyle \int_{0}^{5}f(x)\,dx = \int_{0}^{2}f(x)\,dx + \int_{2}^{5}f(x)\,dx \)
\( \displaystyle 10 = 6 + \int_{2}^{5}f(x)\,dx,\quad\text{so}\quad\int_{2}^{5}f(x)\,dx = 4 \)
Interchanging the limits changes the sign:
\( \displaystyle \int_{5}^{2}f(x)\,dx = -4 \)
By linearity:
\( \displaystyle \int_{2}^{5}\left[2f(x)-g(x)\right]dx = 2(4) - 3 = 5 \)
\( \displaystyle \int_{0}^{2}3f(x)\,dx = 3(6) = 18 \)
So \(\int_{2}^{5} f = 4\), \(\int_{5}^{2} f = -4\), \(\int_{2}^{5}(2f-g) = 5\) and \(\int_{0}^{2}3f = 18\).
Checking the statements
(A) \(\int_{2}^{5}f(x)\,dx = 4\) — Correct. \(10 - 6 = 4\).
(B) \(\int_{5}^{2}f(x)\,dx = 4\) — Incorrect. Interchanging the limits gives \(-4\).
(C) \(\int_{2}^{5}\left[2f(x)-g(x)\right]dx = 5\) — Correct. \(2(4) - 3 = 5\).
(D) \(\int_{0}^{2}3f(x)\,dx = 2\) — Incorrect. The constant multiplies: \(3(6) = 18\).