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Question: If \(\int x e ^ { x } \cos x d x . = f ( x ) + c\) , then f (x) is equal to...

If ∫xexcos⁡xdx.=f(x)+c\int x e ^ { x } \cos x d x . = f ( x ) + c , then f (x) is equal to

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None of these

Answer
Explanation

Solution

Here I = Rear part of ∫xe(1+i)xdx\int x e ^ { ( 1 + i ) x } d x

= = xe(1+i)x(1+i)−e(1+i)x(1+i)2\frac { x e ^ { ( 1 + i ) x } } { ( 1 + i ) } - \frac { e ^ { ( 1 + i ) x } } { ( 1 + i ) ^ { 2 } }

= e(1+i)x {x(1+i)−1(1+i)2}\left\{ \frac { \mathrm { x } ( 1 + \mathrm { i } ) - 1 } { ( 1 + \mathrm { i } ) ^ { 2 } } \right\}

= ex [cos x + i sin x] [(x−1)+ix1+2i−1]\left[ \frac { ( x - 1 ) + i x } { 1 + 2 i - 1 } \right]

= = ex−2\frac { \mathrm { e } ^ { \mathrm { x } } } { - 2 } [(1-x) sin x – x cos x] + c = ex2\frac { \mathrm { e } ^ { \mathrm { x } } } { 2 } [x cos x + (x – 1) sin x] + c