Let \(I = \int_{0}^{\frac{\pi}{2}}\displaystyle e^{x}\sin x\,dx\) and \(J = \int_{0}^{\frac{\pi}{2}}\displaystyle e^{x}\cos x\,dx\). Which of the following statements are correct?
(A) \(I + J = e^{\pi/2}\)
(B) \(I - J = 1\)
(C) \(J = \displaystyle\frac{e^{\pi/2}+1}{2}\)
(D) \(IJ = \displaystyle\frac{e^{\pi}-1}{4}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
Since \(\frac{d}{dx}\left(e^{x}\sin x\right) = e^{x}(\sin x + \cos x)\),
\( \displaystyle I + J = \int_{0}^{\pi/2}e^{x}(\sin x+\cos x)\,dx = \left[e^{x}\sin x\right]_{0}^{\pi/2} = e^{\pi/2} \)
Since \(\frac{d}{dx}\left(-e^{x}\cos x\right) = e^{x}(\sin x - \cos x)\),
\( \displaystyle I - J = \left[-e^{x}\cos x\right]_{0}^{\pi/2} = 0 - (-1) = 1 \)
Adding and subtracting these two results:
\( \displaystyle I = \frac{e^{\pi/2}+1}{2} \)
\( \displaystyle J = \frac{e^{\pi/2}-1}{2} \)
\( \displaystyle IJ = \frac{\left(e^{\pi/2}\right)^{2}-1}{4} = \frac{e^{\pi}-1}{4} \)
So \(I + J = e^{\pi/2}\), \(I - J = 1\), \(J = \frac{e^{\pi/2}-1}{2}\) and \(IJ = \frac{e^{\pi}-1}{4}\).
Checking the statements
(A) \(I + J = e^{\pi/2}\) — Correct. \(e^{\pi/2}\sin\frac{\pi}{2} - e^{0}\sin 0 = e^{\pi/2}\).
(B) \(I - J = 1\) — Correct. \(-e^{\pi/2}\cos\frac{\pi}{2} + e^{0}\cos 0 = 1\).
(C) \(J = \frac{e^{\pi/2}+1}{2}\) — Incorrect. That value is I; \(J = \frac{e^{\pi/2}-1}{2}\).
(D) \(IJ = \frac{e^{\pi}-1}{4}\) — Correct. \(\frac{(e^{\pi/2}+1)(e^{\pi/2}-1)}{4} = \frac{e^{\pi}-1}{4}\).