Which of the following statements are correct?
(A) \(\int_{0}^{1}\displaystyle x^{3}\,dx = \frac{1}{4}\)
(B) \(\int_{1}^{2}\displaystyle\frac{dx}{x^{2}} = \frac{1}{2}\)
(C) \(\int_{0}^{\pi}\displaystyle\cos x\,dx = 2\)
(D) \(\int_{0}^{\log_e 2}\displaystyle e^{x}\,dx = 1\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (D) only
Using the standard antiderivatives:
\( \displaystyle \int_{0}^{1}x^{3}\,dx = \left[\frac{x^{4}}{4}\right]_{0}^{1} = \frac{1}{4} \)
\( \displaystyle \int_{1}^{2}\frac{dx}{x^{2}} = \left[-\frac{1}{x}\right]_{1}^{2} = -\frac{1}{2} + 1 = \frac{1}{2} \)
\( \displaystyle \int_{0}^{\pi}\cos x\,dx = \left[\sin x\right]_{0}^{\pi} = 0 - 0 = 0 \)
\( \displaystyle \int_{0}^{\log_e 2}e^{x}\,dx = e^{\log_e 2} - e^{0} = 2 - 1 = 1 \)
So the values are (A) \(\frac{1}{4}\), (B) \(\frac{1}{2}\), (C) 0 and (D) 1.
Checking the statements
(A) \(\int_{0}^{1}x^{3}\,dx = \frac{1}{4}\) — Correct. \(\frac{1^{4}}{4} - 0 = \frac{1}{4}\).
(B) \(\int_{1}^{2}\frac{dx}{x^{2}} = \frac{1}{2}\) — Correct. \(-\frac{1}{2} + 1 = \frac{1}{2}\).
(C) \(\int_{0}^{\pi}\cos x\,dx = 2\) — Incorrect. \(\sin\pi - \sin 0 = 0\).
(D) \(\int_{0}^{\log_e 2}e^{x}\,dx = 1\) — Correct. \(2 - 1 = 1\).