MHT CET · Maths · Definite Integration
\(\int_{0}^{\frac{\pi}{2}}\left(e^{\sin x}-e^{\cos x}\right) d x=\)
- A \(\frac{1}{2}\)
- B \(0\)
- C \(1\)
- D \(\frac{\pi}{4}\)
Answer & Solution
Correct Answer
(B) \(0\)
Step-by-step Solution
Detailed explanation
\( \text { Let} I =\int_{0}^{\frac{\pi}{2}}\left(e^{\sin x}-e^{\cos x}\right) d x ....(1)\)
\( =\int_{0}^{\pi / 2} e^{\sin \left(\frac{\pi}{2}-x\right)}-e^{\cos \left(\frac{\pi}{2}-x\right)} \mathrm{dx} \)
\( =\int_{0}^{\pi / 2} e^{\cos x}-e^{\sin x} d x....(2)\)
Adding equation (1) \& (2) we get
\(
\begin{array}{l}
2 I=\int_{0}^{\pi / 2}\left(e^{\sin x}-e^{\cos x}-e^{\cos x}-e^{\sin x}\right) d x \\
2 I=0 \Rightarrow I=0
\end{array}
\)
\( =\int_{0}^{\pi / 2} e^{\sin \left(\frac{\pi}{2}-x\right)}-e^{\cos \left(\frac{\pi}{2}-x\right)} \mathrm{dx} \)
\( =\int_{0}^{\pi / 2} e^{\cos x}-e^{\sin x} d x....(2)\)
Adding equation (1) \& (2) we get
\(
\begin{array}{l}
2 I=\int_{0}^{\pi / 2}\left(e^{\sin x}-e^{\cos x}-e^{\cos x}-e^{\sin x}\right) d x \\
2 I=0 \Rightarrow I=0
\end{array}
\)
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