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Question: If \(\int_{}^{}{e^{x}\sin xdx = \frac{1}{2}e^{x}.a + c,}\) then \(a =\)...

If ∫exsin⁡xdx=12ex.a+c,\int_{}^{}{e^{x}\sin xdx = \frac{1}{2}e^{x}.a + c,} then a=a =

A

sin⁡x−cos⁡x\sin x - \cos x

B

cos⁡x−sin⁡x\cos x - \sin x

C

tan⁡x+c\tan x + c

D

None of these

Answer

sin⁡x−cos⁡x\sin x - \cos x

Explanation

Solution

∫exsin⁡xdx=ex12+12(1.sin⁡x−1.cos⁡x)+c=ex2(sin⁡x−cos⁡x)+c\int_{}^{}{e^{x}\sin xdx = \frac{e^{x}}{1^{2} + 1^{2}}(1.\sin x - 1.\cos x) + c} = \frac{e^{x}}{2}(\sin x - \cos x) + c

Clearly a=(sin⁡x−cos⁡x)a = (\sin x - \cos x)