Home > Calculus > Strategies to Test an Infinite Series for Convergence

How do you test the improper integral #int (x(1+x^2)^-2)dx# from #[0,oo)# and evaluate if possible?

Answer 1

Scarlett Allison

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

Evaluate first the indefinite integral:

#F(x) = int x(1+x^2)^-2dx#

by substituting #(1+x^2) = t#, so that #dt = 2xdx#:

#int x(1+x^2)^-2dx = 1/2 int t^-2dt =-1/(2t)+C =-1/(2(1+x^2))+C#

Now we have:

#F(0) = -1/2+C#

#lim_(x->oo) F(x) = C#

So the indefinite integral is convergent and we have:

#int_0^oo x(1+x^2)^-2dx = (lim_(x->oo) F(x)) - F(0) = 1/2#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Isaiah Allen

To test the improper integral ∫(x(1+x^2)^-2)dx from 0 to infinity and evaluate if possible, you can follow these steps:

Determine the convergence of the integral by analyzing its behavior as x approaches infinity.
Find any singular points within the interval of integration.
Apply appropriate techniques, such as integration by substitution or partial fractions, if necessary.
If the integral converges, evaluate it using the fundamental theorem of calculus or other applicable methods.

For this specific integral, you can follow these steps to evaluate it.

By signing up, you agree to our Terms of Service and Privacy Policy

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.