Sr Examen
Integral de (1/(x+1)^3)*arctg(1/((x+1)^2)) dx
Solución
Ha introducido
[src]
2
/
|
| / 1 \
| atan|--------|
| | 2|
| \(x + 1) /
| -------------- dx
| 3
| (x + 1)
|
/
___
-1 + \/ 3
$$\int\limits_{-1 + \sqrt{3}}^{2} \frac{\operatorname{atan}{\left(\frac{1}{\left(x + 1\right)^{2}} \right)}}{\left(x + 1\right)^{3}}\, dx$$
Integral(atan(1/((x + 1)^2))/(x + 1)^3, (x, -1 + sqrt(3), 2))
Gráfica
Respuesta
[src]
/ 1 \
2*atan|----------------------------|
/ 4 3 2\ | 2 | 2 / 4 3 2\ / 4 3 2\ 2
| / ___\ ___ / ___\ / ___\ | | / ___\ ___| / ___\ / ___\ | / ___\ ___ / ___\ / ___\ | / ___\ | / ___\ ___ / ___\ / ___\ | / ___\ / ___\ / ___\ / ___\
atan(1/9) log(82) log\-2 + \-1 + \/ 3 / + 4*\/ 3 + 4*\-1 + \/ 3 / + 6*\-1 + \/ 3 / / \-1 + \-1 + \/ 3 / + 2*\/ 3 / 4*log\\/ 3 / \-1 + \/ 3 / *log\-2 + \-1 + \/ 3 / + 4*\/ 3 + 4*\-1 + \/ 3 / + 6*\-1 + \/ 3 / / 2*\-1 + \/ 3 /*log\-2 + \-1 + \/ 3 / + 4*\/ 3 + 4*\-1 + \/ 3 / + 6*\-1 + \/ 3 / / 4*\-1 + \/ 3 / *log\\/ 3 / 8*\-1 + \/ 3 /*log\\/ 3 /
-log(3) - --------- + ------- - --------------------------------------------------------------------- + ------------------------------------ + ------------------------------ - ----------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------ + ------------------------------ + ------------------------------
18 4 2 2 2 2 2 2 2
/ ___\ ___ / ___\ ___ / ___\ ___ / ___\ ___ / ___\ ___ / ___\ ___ / ___\ ___
-4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3
$$- \log{\left(3 \right)} - \frac{2 \left(-1 + \sqrt{3}\right) \log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\left(-1 + \sqrt{3}\right)^{2} \log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\operatorname{atan}{\left(\frac{1}{9} \right)}}{18} + \frac{2 \operatorname{atan}{\left(\frac{1}{-1 + \left(-1 + \sqrt{3}\right)^{2} + 2 \sqrt{3}} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{4 \left(-1 + \sqrt{3}\right)^{2} \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{4 \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{8 \left(-1 + \sqrt{3}\right) \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{\log{\left(82 \right)}}{4}$$
=
=
/ 1 \
2*atan|----------------------------|
/ 4 3 2\ | 2 | 2 / 4 3 2\ / 4 3 2\ 2
| / ___\ ___ / ___\ / ___\ | | / ___\ ___| / ___\ / ___\ | / ___\ ___ / ___\ / ___\ | / ___\ | / ___\ ___ / ___\ / ___\ | / ___\ / ___\ / ___\ / ___\
atan(1/9) log(82) log\-2 + \-1 + \/ 3 / + 4*\/ 3 + 4*\-1 + \/ 3 / + 6*\-1 + \/ 3 / / \-1 + \-1 + \/ 3 / + 2*\/ 3 / 4*log\\/ 3 / \-1 + \/ 3 / *log\-2 + \-1 + \/ 3 / + 4*\/ 3 + 4*\-1 + \/ 3 / + 6*\-1 + \/ 3 / / 2*\-1 + \/ 3 /*log\-2 + \-1 + \/ 3 / + 4*\/ 3 + 4*\-1 + \/ 3 / + 6*\-1 + \/ 3 / / 4*\-1 + \/ 3 / *log\\/ 3 / 8*\-1 + \/ 3 /*log\\/ 3 /
-log(3) - --------- + ------- - --------------------------------------------------------------------- + ------------------------------------ + ------------------------------ - ----------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------ + ------------------------------ + ------------------------------
18 4 2 2 2 2 2 2 2
/ ___\ ___ / ___\ ___ / ___\ ___ / ___\ ___ / ___\ ___ / ___\ ___ / ___\ ___
-4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3 -4 + 4*\-1 + \/ 3 / + 8*\/ 3
$$- \log{\left(3 \right)} - \frac{2 \left(-1 + \sqrt{3}\right) \log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\left(-1 + \sqrt{3}\right)^{2} \log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\operatorname{atan}{\left(\frac{1}{9} \right)}}{18} + \frac{2 \operatorname{atan}{\left(\frac{1}{-1 + \left(-1 + \sqrt{3}\right)^{2} + 2 \sqrt{3}} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{4 \left(-1 + \sqrt{3}\right)^{2} \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{4 \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{8 \left(-1 + \sqrt{3}\right) \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{\log{\left(82 \right)}}{4}$$
-log(3) - atan(1/9)/18 + log(82)/4 - log(-2 + (-1 + sqrt(3))^4 + 4*sqrt(3) + 4*(-1 + sqrt(3))^3 + 6*(-1 + sqrt(3))^2)/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) + 2*atan(1/(-1 + (-1 + sqrt(3))^2 + 2*sqrt(3)))/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) + 4*log(sqrt(3))/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) - (-1 + sqrt(3))^2*log(-2 + (-1 + sqrt(3))^4 + 4*sqrt(3) + 4*(-1 + sqrt(3))^3 + 6*(-1 + sqrt(3))^2)/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) - 2*(-1 + sqrt(3))*log(-2 + (-1 + sqrt(3))^4 + 4*sqrt(3) + 4*(-1 + sqrt(3))^3 + 6*(-1 + sqrt(3))^2)/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) + 4*(-1 + sqrt(3))^2*log(sqrt(3))/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) + 8*(-1 + sqrt(3))*log(sqrt(3))/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.