Question:
What is \(\int_0^{\pi} e^x \sin x\,dx\) equal to?
\(\int_0^{\pi} e^x \sin x\,dx\) किसके बराबर है?
Options:
- \(\dfrac{e^\pi + 1}{4}\)
- \(\dfrac{e^\pi – 1}{2}\)
- \(e^\pi + 1\)
- \(\dfrac{e^\pi + 1}{2}\)
- \(\dfrac{e^\pi + 1}{4}\)
- \(\dfrac{e^\pi – 1}{2}\)
- \(e^\pi + 1\)
- \(\dfrac{e^\pi + 1}{2}\)
Correct Answer: d)
Explanation:
Using the standard formula \(\int e^{ax}\sin(bx)\,dx = \dfrac{e^{ax}(a\sin bx – b\cos bx)}{a^2+b^2}\):
\(\int_0^\pi e^x \sin x\,dx = \left[\dfrac{e^x(\sin x – \cos x)}{2}\right]_0^\pi = \dfrac{e^\pi(0 – (-1)) – e^0(0 – 1)}{2} = \dfrac{e^\pi + 1}{2}\).
मानक सूत्र से: \(\left[\dfrac{e^x(\sin x – \cos x)}{2}\right]_0^\pi = \dfrac{e^\pi + 1}{2}\)।