Calc1.50

From Example Problems
Revision as of 14:15, 27 February 2006 by Geoff (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

\int {\frac  {2x}{x^{2}+1}}\,dx\,

To find this integral, start by making a substitution.

Let u=x^{2}+1\,

Then du=2xdx\,

Now solve for dx\, to get

dx={\frac  {du}{2x}}\,

Now substitute these functions into the integral.

\int {\frac  {2x}{x^{2}+1}}\,dx=\int {\frac  {2x}{u}}{\frac  {du}{2x}}=\int {\frac  {du}{u}}\,

Now, this integral is of a form of which we already know the technique.

\int {\frac  {du}{u}}=\ln |u|+C\,

The final step is to undo the substition. Plug x^{2}+1\, back in wherever there is a u\,. Also, note the absolute value bars can be dropped because x^{2}+1\, is always positive. So our answer is

\ln(x^{2}+1)+C\,


Main Page : Calculus