_ ^ e^x x( x + 2 x) ;dx = - Vidyadip

MCQ

$\int_{}^{} {{e^x}\sin x(\sin x + 2\cos x)} \;dx = $

✓
${e^x}{\sin ^2}x + c$
B
${e^x}\sin x + c$
C
${e^x}\sin 2x + c$
D
None of these

✓

Answer

Correct option: A.

${e^x}{\sin ^2}x + c$

a
(a)$\int_{}^{} {{e^x}\sin x(\sin x + 2\cos x)dx} $
$ = \int_{}^{} {{e^x}{{\sin }^2}x\,dx} + \int_{}^{} {{e^x}2\sin x\,\cos xdx} $
$ = \int_{}^{} {{e^x}{{\sin }^2}x\,dx} + \int_{}^{} {{e^x}\sin 2x\,dx} $
$ = {e^x}{\sin ^2}x - \int_{}^{} {{e^x}\sin 2x\,dx} + \int_{}^{} {{e^x}\sin 2x\,dx\, + c} $
$ = {e^x}{\sin ^2}x + c.$
Aliter : $\int_{}^{} {{e^x}({{\sin }^2}x + \sin 2x)dx = {e^x}{{\sin }^2}x + c.} $

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