Pages

News: Currently the LaTeX and hidden solutions on this blog do not work on Google Reader.
Email me if you have suggestions on how to improve this blog!

Saturday 13 August 2011

Math GRE - #12

A simple one today:
What is the value of the integral: \[\int_0^1{\frac{x}{1+x^2}}dx?\]
  • $1$
  • $\frac{\pi}{4}$
  • $\arctan\frac{\sqrt{2}}{2}$
  • $\log 2$
  • $\log\sqrt{2}$

Solution :

We can use the substitution $u=1+x^2\implies du = 2x\,dx$. With this substitution, the original integral becomes: \[\int_0^1{\frac{x}{1+x^2}}dx=\frac{1}{2}\int_1^2{\frac{1}{u}}du\]
Upon computing this integral, we obtain: \[\frac{1}{2}\int_1^2{\frac{1}{u}}du=\frac{1}{2}\left. \log{u}\right|_1^2=\frac{1}{2}\log{2}=\log{\sqrt{2}}.\]

0 comments:

Post a Comment

This webpage is LaTeX enabled. To type in-line formulae, type your stuff between two '$'. To type centred formulae, type '\[' at the beginning of your formula and '\]' at the end.