Hello, I have to integrate the following by substitution, but I am having trouble... any help would be appreciated...
$\displaystyle \int x \log (x^2 + 1)dx $. thanks
Hello, slevvio!
This cannot be integrated by substitution alone.I have to integrate the following by substitution. .?
. . . $\displaystyle \int x \log (x^2 + 1)dx $
We must integrate by parts . . .
. . $\displaystyle \begin{array}{ccccccc}u & = & \ln(x^2+1) & \quad & dv & = & x\,dx \\ du & = & \frac{2x}{x^2+1}\,dx & \quad & v & = & \frac{1}{2}x^2 \end{array}$
Then we have: .$\displaystyle \frac{1}{2}x^2\ln(x^2+1) - 2\int\frac{x^3}{x^2+1}\,dx $
. . . $\displaystyle =\;\frac{1}{2}x^2\ln(x^2+1) - 2\int\left(x - \frac{x}{x^2+1}\right)\,dx$ . . . . etc.