How do you test the improper integral #int absx(x^2+1)^3 dx# from #(oo, oo)# and evaluate if possible?
Then
By signing up, you agree to our Terms of Service and Privacy Policy
The integrand is even:
... and for an even function:
By signing up, you agree to our Terms of Service and Privacy Policy
To test the convergence of the improper integral ( \int_{\infty}^{\infty} x (x^2+1)^{3} dx ) and evaluate if possible, we can use the limit comparison test.

First, we choose a function to compare with. Since the integrand involves ( x ), we choose a function that behaves similarly near infinity. One such function is ( \frac{1}{x^2} ).

We calculate the limit of the ratio of the two functions as ( x ) approaches infinity:
[ \lim_{{x \to \infty}} \frac{\frac{1}{x^2}}{x(x^2+1)^{3}} ]
 Simplifying the expression, we get:
[ \lim_{{x \to \infty}} \frac{1}{x^3(x^2+1)^{3}} ]

Using L'Hôpital's Rule if necessary, we evaluate the limit. After evaluating, if the limit is a positive finite number, then both integrals either converge or diverge.

If the limit is finite and positive, then the original integral converges. Otherwise, it diverges.
Note: If the limit is infinite or undefined, then we may need to choose a different function to compare with.
The evaluation of the integral itself involves integration techniques which can be carried out once convergence is established.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
 How do you test for convergence for #1/((2n+1)!) #?
 How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #11/3+1/9...+(1/3)^n+...#?
 How do you find #lim (1cosx)/x# as #x>0# using l'Hospital's Rule?
 How do you find #lim sqrtx/(x1)# as #x>1^+# using l'Hospital's Rule or otherwise?
 Using the integral test, how do you show whether #sum ln(n)/(n)^2# diverges or converges from n=1 to infinity?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7