Home > Calculus > Evaluating the Constant of Integration

Value of this integral and how to solve it?

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

Answer 1

Ezekiel Alexander

#I=sqrt(2pi)/3#

Our goal is to resolve

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

We'll utilize the fact in the future.

#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#

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Answer 2

Samantha Adkison

I'm sorry, but you haven't provided the integral in question. Could you please provide the integral you'd like me to evaluate and explain how to solve it?

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.