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Question
The value of \[\int\limits_0^{\pi/2} \cos x\ e^{\sin x}\ dx\] is
Options
1
e − 1
0
− 1
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Solution
e − 1
\[Let\, I = \int_0^\frac{\pi}{2} \cos x\ e^{\sin x}\ d x\]
\[Let\ \sin x = t, then\ \cos x dx = dt\]
\[When\ x = 0, t = 0\ and\ x = \frac{\pi}{2}, t = 1\]
\[\text{Therefore the integral becomes}\]
\[I = \int_0^1 e^t dt\]
\[ = \left[ e^t \right]_0^1 \]
\[ = e - 1\]
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