Home
Class 12
MATHS
int(0)^(oo)e^(-x^(2))dx=(sqrtpi)/(2) the...

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

A

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

B

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

C

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

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)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If int_(0)^(oo) e^(-x^(2))dx=sqrt((pi)/(2))"then"int_(0)^(oo) e^(-ax^(2)) dx, a gt0 , s

1. int_(0)^(oo)e^(-x/2)dx

int_(0)^(1)x e^(x^(2)) dx

int_(0)^(4)(x+e^(2x))dx

If int_0^ooe^(-x^2) dx=(sqrtpi)/2 , then int_0^ooe^(-ax^2) dx where a gt 0 is:

int_(0)^(2)e^(x)\ dx

int_(0)^(2) e^(x) dx

int_(0)^(1)e^(2x)e^(e^(x) dx =)

int_(0)^(2) (e^(-1//x))/(x^(2)) dx

int_(-1)^(1)e^(2x)dx