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