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int(0)^(oo)e^(-x^(2))dx=(sqrtpi)/(2) the...

`int_(0)^(oo)e^(-x^(2))dx=(sqrtpi)/(2)` then

A

a) `int_(+0)^(oo)e^(-2x^(2))dx=(sqrtpi)/(2sqrt2)`

B

b) `int_(0)^(oo)xe^(-x^(2))dx=(1)/(2)`

C

c) `int_(0)^(oo)x^(2)e^(-x^(2))dx=(sqrtpi)/(4)`

D

d) `int_(0)^(oo)x^(2)e^(-x^(2))dx=(pi)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
We have `int_(0)^(oo)e^(-x^(2))dx=(sqrtx)/(2)`
For `int_(0)^(oo)e^(-2x^(2))dx,` put `sqrt2x=t,`
`therefore" "int_(0)^(oo)xe^(-x^(2))dx=(1)/(2)int_(0)^(oo)e^(-t)dt=(1)/(2)`
For `int_(0)^(oo)x^(2)e^(-x^(2))dx=int_(0)^(oo)x(xe^(-x^(2)))dx,` integrating by parts,
`={x(-(1)/(2)e^(-x^(2)))}_(0)^(oo)+(1)/(2)int_(0)^(oo)e^(-x^(2))dx=(sqrtpi)/(4)`
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