Question

Calcule as seguintes integrais indefinidas pelo método de integração por partes: a. ∫ x ln x dx

Answer 1

$\large\boxed{\int\ udv=uv-\int\ vdu}$

$u=ln\ x\rightarrow\ du=\frac{1}{x}\ dx\\\\ dv=x\ dx\rightarrow\ v=\frac{x^2}{2}$

$\int\ xlnx\ dx=lnx*\frac{x^2}{2}-\int\ \frac{x^2}{2}*\frac{1}{x}\ dx\\\\\ \int\ xlnx\ dx=lnx*\frac{x^2}{2}-\frac{1}{2}\int\ x\ dx\\\\ \int\ xlnx\ dx=lnx*\frac{x^2}{2}-\frac{x^2}{4}\\\\ \large\boxed{\int\ xlnx\ dx=\frac{x^2}{4}(2lnx-1)+C}$

Answer 2

Boa tarde Carlos!
Solução!

$\int\limits x~ln~x~dx$

$u=ln(x)\\\\\ dv=x \\\\\ du= \dfrac{1}{x}\\\\ v=x$

$\int\limits u.dv=uv- \int\limits v.du$

Substituindo fica assim.

$\int\limits x~ln~x~dx=lnx.x- \int\limits x. \frac{1}{x} dx$

$\int\limits x~ln~x~dx=xlnx- x+c$

Boa tarde!
Bons estudos!