Question

Evaluate the integral or state that it diverges: The integral from 1 to infinity of dx/(2x^2+5)

Answer 1

It looks like the derivative of the arctangent function, only we have to fiddle a little first and then make a substitution:\[\int\limits_{1}^{\infty}\frac{ dx }{ 2x^2+5 }=\int\limits_{1}^{\infty}\frac{ \frac{1}{5}dx }{ 1+\frac{2}{5}x^2 }=\dfrac{1}{5}\int\limits_{1}^{\infty}\frac{ dx }{ 1+(\sqrt{\frac{2}{5}}x)^2 }\]
(ou could replace \(\sqrt{\frac{2}{5}}\) by \(\frac{1}{5}\sqrt{10}\) ).
Now if you make the substitution \(u=\frac{1}{5}\sqrt{10}x\), you will get \(du=\frac{1}{5}\sqrt{10}dx\) and \(dx=\frac{1}{2}\sqrt{10}du\).

Put these values in the integral. It will be the derivative of the arctangent.

Answer 2

Thank you so much. :)

Answer 3

yw!