Question

Value of this integral and how to solve it?

`int_0^prop(1-x^2sin(1/x^2))dx`

Answer 1

`I=sqrt(2pi)/3`

Explanation:

We want to solve

`I=int_0^oo(1-x^2sin(1/x^2))dx`

Later on, we will use the fact

`color(blue)(int_0^oo sin(x^2)dx=sqrt(pi/8)`

Make a substitution `color(red)(u=1/x^2=>du=-2/x^3dx` and `color(red)(x=1/sqrt(u)`

`I=-1/2int_oo^0(u-sin(u))/(u^(5/2))du`

`color(white)(I)=1/2int_0^oo(u-sin(u))/(u^(5/2))du`

Use IBP

`I=-1/3[(u-sin(u))/(u^(3/2))]_0^oo+1/3int_0^oo(1-cos(u))/u^(3/2)du`

But

`color(red)(-1/3[(u-sin(u))/(u^(3/2))]_0^oo=0)larr "Use LHS"`

Thus

`I=1/3int_0^oo(1-cos(u))/u^(3/2)du`

Use IBP

`I=-2/3[(1-cos(u))/sqrt(u)]_0^oo+2/3int_0^oosin(u)/u^(1/2)du`

But

`color(red)(-2/3[(1-cos(u))/sqrt(u)]_0^oo=0)larr "Use LHS"`

Thus

`I=2/3int_0^oosin(u)/u^(1/2)du`

Make a substitution `s=sqrt(u)=>ds=1/(2s)du`

`I=4/3int_0^oosin(s^2)ds=4/3sqrt(pi/8)=sqrt(2pi)/3`