Mathematics

Evaluate: $$\int\limits_0^{\dfrac{\pi }{2}} {\cos x\;{e^{\sin x}}\;{\text{dx}}} $$


SOLUTION
$$\displaystyle I = \int_{0}^{\pi /2} \cos\,x \,e^{\sin\,x}dx $$

Let $$ \sin\,x = t $$           $$ x = \pi /2 $$     $$ t = 1 $$

$$ \Rightarrow \cos\ x \ dx = dt $$    $$ x = 0 $$   $$ t = 0 $$

$$ \displaystyle \Rightarrow  I = \int_{0}^{1} e^{t}dt $$

$$ = [e^{t}]_{0}^{1} = e-1 $$
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Subjective Medium Published on 17th 09, 2020
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