# Value of this integral and how to solve it?

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

##### 1 Answer

#### 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

#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

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