Sr Examen
Integral de 2*x/(x^3+5) dx
Solución
Ha introducido
[src]
oo / | | 2*x | ------ dx | 3 | x + 5 | / 2
$$\int\limits_{2}^{\infty} \frac{2 x}{x^{3} + 5}\, dx$$
Integral((2*x)/(x^3 + 5), (x, 2, oo))
Respuesta (Indefinida)
[src]
/ ___ ___ 2/3\ / ___ 2/3 | \/ 3 2*x*\/ 3 *5 | | 2/3 / 3 ___\ 2/3 / 2/3 2 3 ___\ 2*\/ 3 *5 *atan|- ----- + --------------| | 2*x 2*5 *log\x + \/ 5 / 5 *log\5 + x - x*\/ 5 / \ 3 15 / | ------ dx = C - --------------------- + ----------------------------- + ------------------------------------------- | 3 15 15 15 | x + 5 | /
$$\int \frac{2 x}{x^{3} + 5}\, dx = C - \frac{2 \cdot 5^{\frac{2}{3}} \log{\left(x + \sqrt[3]{5} \right)}}{15} + \frac{5^{\frac{2}{3}} \log{\left(x^{2} - \sqrt[3]{5} x + 5^{\frac{2}{3}} \right)}}{15} + \frac{2 \sqrt{3} \cdot 5^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2 \sqrt{3} \cdot 5^{\frac{2}{3}} x}{15} - \frac{\sqrt{3}}{3} \right)}}{15}$$
Gráfica
Respuesta
[src]
/ / pi*I\ / 5*pi*I\\
| -pi*I | ----| pi*I | ------||
| / 3 ___ pi*I\ ------ | 3 ___ 3 | ---- | 3 ___ 3 ||
| 2/3 | \/ 5 *e | 2/3 3 | \/ 5 *e | 2/3 3 | \/ 5 *e ||
|2*5 *log|1 - -----------| 2*5 *e *log|1 - -----------| 2*5 *e *log|1 - -------------||
| \ 2 / \ 2 / \ 2 /|
|--------------------------- - ----------------------------------- - -----------------------------------|*Gamma(1/3)
\ 15 15 15 /
--------------------------------------------------------------------------------------------------------------------
3*Gamma(4/3)
$$\frac{\left(- \frac{2 \cdot 5^{\frac{2}{3}} e^{\frac{i \pi}{3}} \log{\left(- \frac{\sqrt[3]{5} e^{\frac{5 i \pi}{3}}}{2} + 1 \right)}}{15} + \frac{2 \cdot 5^{\frac{2}{3}} \log{\left(- \frac{\sqrt[3]{5} e^{i \pi}}{2} + 1 \right)}}{15} - \frac{2 \cdot 5^{\frac{2}{3}} e^{- \frac{i \pi}{3}} \log{\left(1 - \frac{\sqrt[3]{5} e^{\frac{i \pi}{3}}}{2} \right)}}{15}\right) \Gamma\left(\frac{1}{3}\right)}{3 \Gamma\left(\frac{4}{3}\right)}$$
=
=
/ / pi*I\ / 5*pi*I\\
| -pi*I | ----| pi*I | ------||
| / 3 ___ pi*I\ ------ | 3 ___ 3 | ---- | 3 ___ 3 ||
| 2/3 | \/ 5 *e | 2/3 3 | \/ 5 *e | 2/3 3 | \/ 5 *e ||
|2*5 *log|1 - -----------| 2*5 *e *log|1 - -----------| 2*5 *e *log|1 - -------------||
| \ 2 / \ 2 / \ 2 /|
|--------------------------- - ----------------------------------- - -----------------------------------|*Gamma(1/3)
\ 15 15 15 /
--------------------------------------------------------------------------------------------------------------------
3*Gamma(4/3)
$$\frac{\left(- \frac{2 \cdot 5^{\frac{2}{3}} e^{\frac{i \pi}{3}} \log{\left(- \frac{\sqrt[3]{5} e^{\frac{5 i \pi}{3}}}{2} + 1 \right)}}{15} + \frac{2 \cdot 5^{\frac{2}{3}} \log{\left(- \frac{\sqrt[3]{5} e^{i \pi}}{2} + 1 \right)}}{15} - \frac{2 \cdot 5^{\frac{2}{3}} e^{- \frac{i \pi}{3}} \log{\left(1 - \frac{\sqrt[3]{5} e^{\frac{i \pi}{3}}}{2} \right)}}{15}\right) \Gamma\left(\frac{1}{3}\right)}{3 \Gamma\left(\frac{4}{3}\right)}$$
(2*5^(2/3)*log(1 - 5^(1/3)*exp_polar(pi*i)/2)/15 - 2*5^(2/3)*exp(-pi*i/3)*log(1 - 5^(1/3)*exp_polar(pi*i/3)/2)/15 - 2*5^(2/3)*exp(pi*i/3)*log(1 - 5^(1/3)*exp_polar(5*pi*i/3)/2)/15)*gamma(1/3)/(3*gamma(4/3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.