Sr Examen
Integral de dx/(x^2*sqrt(x^2)-5) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | ____ | 2 / 2 | x *\/ x - 5 | / 0
$$\int\limits_{0}^{1} \frac{1}{x^{2} \sqrt{x^{2}} - 5}\, dx$$
Integral(1/(x^2*sqrt(x^2) - 5), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ ____\
| ___ ___ 2/3 / 2 |
/ / ____\ / ____ \ ___ 3 ___ |\/ 3 2*\/ 3 *5 *\/ x |
| 3 ___ | 2/3 2 3 ___ / 2 | 3 ___ | / 2 3 ___| \/ 3 *\/ 5 *atan|----- + --------------------|
| 1 \/ 5 *log\5 + x + \/ 5 *\/ x / \/ 5 *log\\/ x - \/ 5 / \ 3 15 /
| -------------- dx = C - ------------------------------------ + -------------------------- - ----------------------------------------------
| ____ 30 15 15
| 2 / 2
| x *\/ x - 5
|
/
$$\int \frac{1}{x^{2} \sqrt{x^{2}} - 5}\, dx = C + \frac{\sqrt[3]{5} \log{\left(\sqrt{x^{2}} - \sqrt[3]{5} \right)}}{15} - \frac{\sqrt[3]{5} \log{\left(x^{2} + \sqrt[3]{5} \sqrt{x^{2}} + 5^{\frac{2}{3}} \right)}}{30} - \frac{\sqrt{3} \sqrt[3]{5} \operatorname{atan}{\left(\frac{2 \sqrt{3} \cdot 5^{\frac{2}{3}} \sqrt{x^{2}}}{15} + \frac{\sqrt{3}}{3} \right)}}{15}$$
Gráfica
Respuesta
[src]
/ ___ ___ 2/3\
___ 3 ___ |\/ 3 2*\/ 3 *5 |
3 ___ / /3 ___\\ 3 ___ / 3 ___ 2/3\ 3 ___ / / 3 ___\\ 3 ___ / 2/3\ \/ 3 *\/ 5 *atan|----- + ------------| ___ 3 ___
\/ 5 *\pi*I + log\\/ 5 // \/ 5 *log\1 + \/ 5 + 5 / \/ 5 *\pi*I + log\-1 + \/ 5 // \/ 5 *log\5 / \ 3 15 / pi*\/ 3 *\/ 5
- ------------------------- - --------------------------- + ------------------------------ + --------------- - -------------------------------------- + --------------
15 30 15 30 15 90
$$- \frac{\sqrt{3} \sqrt[3]{5} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \cdot 5^{\frac{2}{3}}}{15} \right)}}{15} - \frac{\sqrt[3]{5} \log{\left(1 + \sqrt[3]{5} + 5^{\frac{2}{3}} \right)}}{30} + \frac{\sqrt[3]{5} \log{\left(5^{\frac{2}{3}} \right)}}{30} + \frac{\sqrt{3} \sqrt[3]{5} \pi}{90} - \frac{\sqrt[3]{5} \left(\log{\left(\sqrt[3]{5} \right)} + i \pi\right)}{15} + \frac{\sqrt[3]{5} \left(\log{\left(-1 + \sqrt[3]{5} \right)} + i \pi\right)}{15}$$
=
=
/ ___ ___ 2/3\
___ 3 ___ |\/ 3 2*\/ 3 *5 |
3 ___ / /3 ___\\ 3 ___ / 3 ___ 2/3\ 3 ___ / / 3 ___\\ 3 ___ / 2/3\ \/ 3 *\/ 5 *atan|----- + ------------| ___ 3 ___
\/ 5 *\pi*I + log\\/ 5 // \/ 5 *log\1 + \/ 5 + 5 / \/ 5 *\pi*I + log\-1 + \/ 5 // \/ 5 *log\5 / \ 3 15 / pi*\/ 3 *\/ 5
- ------------------------- - --------------------------- + ------------------------------ + --------------- - -------------------------------------- + --------------
15 30 15 30 15 90
$$- \frac{\sqrt{3} \sqrt[3]{5} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \cdot 5^{\frac{2}{3}}}{15} \right)}}{15} - \frac{\sqrt[3]{5} \log{\left(1 + \sqrt[3]{5} + 5^{\frac{2}{3}} \right)}}{30} + \frac{\sqrt[3]{5} \log{\left(5^{\frac{2}{3}} \right)}}{30} + \frac{\sqrt{3} \sqrt[3]{5} \pi}{90} - \frac{\sqrt[3]{5} \left(\log{\left(\sqrt[3]{5} \right)} + i \pi\right)}{15} + \frac{\sqrt[3]{5} \left(\log{\left(-1 + \sqrt[3]{5} \right)} + i \pi\right)}{15}$$
-5^(1/3)*(pi*i + log(5^(1/3)))/15 - 5^(1/3)*log(1 + 5^(1/3) + 5^(2/3))/30 + 5^(1/3)*(pi*i + log(-1 + 5^(1/3)))/15 + 5^(1/3)*log(5^(2/3))/30 - sqrt(3)*5^(1/3)*atan(sqrt(3)/3 + 2*sqrt(3)*5^(2/3)/15)/15 + pi*sqrt(3)*5^(1/3)/90
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.