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

Question

William Abdalla · Accepted Answer

See below.

#(2x)/(sqrt(x^2+1))# is an odd function so #int_(-a)^a (2x)/(sqrt(x^2+1)) dx = 0#

Emma Aldridge · Answer

undefined To test the improper integral $ \int_{-\infty}^{\infty} \frac{2x}{\sqrt{x^2+1}} \, dx $, first, split the integral into two separate integrals from negative infinity to zero and from zero to positive infinity. Then apply substitution or use limits to evaluate each integral.

Let's denote the first integral as $ I_1 $ from negative infinity to zero, and the second integral as $ I_2 $ from zero to positive infinity.

For $ I_1 $, perform the substitution $ u = -x^2 - 1 $, and for $ I_2 $, let $ u = x^2 + 1 $.

After substitution, both integrals should be expressed in terms of $ u $. Then, find the limits of integration as $ x $ approaches negative infinity and positive infinity.

Finally, evaluate each integral using these limits. If both integrals converge (i.e., yield finite values), then the original integral converges. If not, it diverges.

In this case, both integrals converge, thus the original integral converges.