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Question: \(\int _ { - 10 } ^ { 10 } x e ^ { x \left[ x + \frac { 1 } { 2 } \right] }\) dx is equal to ([x] de...

∫−1010xex[x+12]\int _ { - 10 } ^ { 10 } x e ^ { x \left[ x + \frac { 1 } { 2 } \right] } dx is equal to ([x] denotes the integral part of x)

A

0

B

1

C

2

D

3

Answer

0

Explanation

Solution

I = dx = ∫−1010(−x)⋅e−x([−x+12])\int _ { - 10 } ^ { 10 } ( - x ) \cdot e ^ { - x \left( \left[ - x + \frac { 1 } { 2 } \right] \right) } dx

Now [−x+12]\left[ - \mathrm { x } + \frac { 1 } { 2 } \right] = [−(x+12)+1]\left[ - \left( x + \frac { 1 } { 2 } \right) + 1 \right] = – [x+12]\left[ \mathrm { x } + \frac { 1 } { 2 } \right]

If x ¹ odd multiple of 12\frac { 1 } { 2 } .

I = = – I ̃ I = 0.