Sr Examen
Integral de (x^2)/(x^4+1) dx
Solución
Ha introducido
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9/10 / | | 2 | x | ------ dx | 4 | x + 1 | / -9/10
$$\int\limits_{- \frac{9}{10}}^{\frac{9}{10}} \frac{x^{2}}{x^{4} + 1}\, dx$$
Integral(x^2/(x^4 + 1), (x, -9/10, 9/10))
Respuesta (Indefinida)
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/ | | 2 ___ / 2 ___\ ___ / ___\ ___ / ___\ ___ / 2 ___\ | x \/ 2 *log\1 + x + x*\/ 2 / \/ 2 *atan\1 + x*\/ 2 / \/ 2 *atan\-1 + x*\/ 2 / \/ 2 *log\1 + x - x*\/ 2 / | ------ dx = C - --------------------------- + ----------------------- + ------------------------ + --------------------------- | 4 8 4 4 8 | x + 1 | /
$$\int \frac{x^{2}}{x^{4} + 1}\, dx = C + \frac{\sqrt{2} \log{\left(x^{2} - \sqrt{2} x + 1 \right)}}{8} - \frac{\sqrt{2} \log{\left(x^{2} + \sqrt{2} x + 1 \right)}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x - 1 \right)}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} x + 1 \right)}}{4}$$
Gráfica
Respuesta
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/ ___\ / ___\ / ___\ / ___\
___ | 9*\/ 2 | ___ | 9*\/ 2 | ___ |181 9*\/ 2 | ___ |181 9*\/ 2 |
\/ 2 *atan|1 + -------| \/ 2 *atan|1 - -------| \/ 2 *log|--- + -------| \/ 2 *log|--- - -------|
\ 10 / \ 10 / \100 10 / \100 10 /
----------------------- - ----------------------- - ------------------------ + ------------------------
2 2 4 4
$$- \frac{\sqrt{2} \log{\left(\frac{9 \sqrt{2}}{10} + \frac{181}{100} \right)}}{4} + \frac{\sqrt{2} \log{\left(\frac{181}{100} - \frac{9 \sqrt{2}}{10} \right)}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left(1 - \frac{9 \sqrt{2}}{10} \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(1 + \frac{9 \sqrt{2}}{10} \right)}}{2}$$
=
=
/ ___\ / ___\ / ___\ / ___\
___ | 9*\/ 2 | ___ | 9*\/ 2 | ___ |181 9*\/ 2 | ___ |181 9*\/ 2 |
\/ 2 *atan|1 + -------| \/ 2 *atan|1 - -------| \/ 2 *log|--- + -------| \/ 2 *log|--- - -------|
\ 10 / \ 10 / \100 10 / \100 10 /
----------------------- - ----------------------- - ------------------------ + ------------------------
2 2 4 4
$$- \frac{\sqrt{2} \log{\left(\frac{9 \sqrt{2}}{10} + \frac{181}{100} \right)}}{4} + \frac{\sqrt{2} \log{\left(\frac{181}{100} - \frac{9 \sqrt{2}}{10} \right)}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left(1 - \frac{9 \sqrt{2}}{10} \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(1 + \frac{9 \sqrt{2}}{10} \right)}}{2}$$
sqrt(2)*atan(1 + 9*sqrt(2)/10)/2 - sqrt(2)*atan(1 - 9*sqrt(2)/10)/2 - sqrt(2)*log(181/100 + 9*sqrt(2)/10)/4 + sqrt(2)*log(181/100 - 9*sqrt(2)/10)/4
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.