Sr Examen
Integral de (x*2)/(pi*(1+x^2))*(x*arctg(x)-(1/2)*ln(x^2+1)) dx
Solución
Ha introducido
[src]
1 / | | / / 2 \\ | x*2 | log\x + 1/| | -----------*|x*atan(x) - -----------| dx | / 2\ \ 2 / | pi*\1 + x / | / 0
$$\int\limits_{0}^{1} \frac{2 x}{\pi \left(x^{2} + 1\right)} \left(x \operatorname{atan}{\left(x \right)} - \frac{\log{\left(x^{2} + 1 \right)}}{2}\right)\, dx$$
Integral(((x*2)/((pi*(1 + x^2))))*(x*atan(x) - log(x^2 + 1)/2), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ / / 2\ 2 \ | |log\1 + x / atan (x)| | / / 2 \\ 2*|----------- - --------| 2/ 2\ | x*2 | log\x + 1/| \ 2 2 / log \1 + x / 2*(x - atan(x))*atan(x) | -----------*|x*atan(x) - -----------| dx = C - -------------------------- - ------------ + ----------------------- | / 2\ \ 2 / pi 4*pi pi | pi*\1 + x / | /
$$\int \frac{2 x}{\pi \left(x^{2} + 1\right)} \left(x \operatorname{atan}{\left(x \right)} - \frac{\log{\left(x^{2} + 1 \right)}}{2}\right)\, dx = C + \frac{2 \left(x - \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}{\left(x \right)}}{\pi} - \frac{2 \left(\frac{\log{\left(x^{2} + 1 \right)}}{2} - \frac{\operatorname{atan}^{2}{\left(x \right)}}{2}\right)}{\pi} - \frac{\log{\left(x^{2} + 1 \right)}^{2}}{4 \pi}$$
Gráfica
Respuesta
[src]
/ 2 2 \
| log(2) log (2) pi pi|
2*|- ------ - ------- - --- + --|
\ 2 8 32 4 /
---------------------------------
pi
$$\frac{2 \left(- \frac{\log{\left(2 \right)}}{2} - \frac{\pi^{2}}{32} - \frac{\log{\left(2 \right)}^{2}}{8} + \frac{\pi}{4}\right)}{\pi}$$
=
=
/ 2 2 \
| log(2) log (2) pi pi|
2*|- ------ - ------- - --- + --|
\ 2 8 32 4 /
---------------------------------
pi
$$\frac{2 \left(- \frac{\log{\left(2 \right)}}{2} - \frac{\pi^{2}}{32} - \frac{\log{\left(2 \right)}^{2}}{8} + \frac{\pi}{4}\right)}{\pi}$$
2*(-log(2)/2 - log(2)^2/8 - pi^2/32 + pi/4)/pi
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.