If _ ^ e^x x ;dx = 1 2 e^x ;. ;a + c , then a = - Vidyadip

MCQ

If $\int_{}^{} {{e^x}\sin x\;dx = \frac{1}{2}{e^x}\;.\;a + c} $, then $a = $

✓
$\sin x - \cos x$
B
$\cos x - \sin x$
C
$ - \cos x - \sin x$
D
$\cos x + \sin x$

✓

Answer

Correct option: A.

$\sin x - \cos x$

a
(a) Given that $\int_{}^{} {{e^x}\sin x\,dx} = \frac{1}{2}{e^x}a + c$ ...$(i)$

Let $I = \int_{}^{} {{e^x}\sin x\,dx} = - {e^x}\cos x + \int_{}^{} {{e^x}\cos x\,dx + c} $

$ = - {e^x}\cos x + {e^x}\sin x - \int_{}^{} {{e^x}\sin x\,dx + c} $

$ \Rightarrow 2I = {e^x}( - \cos x + \sin x) + c$. Now from $(i),$

we get $\frac{1}{2}{e^x}a = \frac{1}{2}{e^x}(\sin x - \cos x) \Rightarrow a = \sin x - \cos x.$

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