Sr Examen
Integral de dx/(x*(sqrt(2)*sqrt(x))+1) dx
Solución
Ha introducido
[src]
1 / | | 1 | ----------------- dx | ___ ___ | x*\/ 2 *\/ x + 1 | / 0
$$\int\limits_{0}^{1} \frac{1}{x \sqrt{2} \sqrt{x} + 1}\, dx$$
Integral(1/(x*(sqrt(2)*sqrt(x)) + 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ 5/6\ / ___ 6 ___ ___ ___\ / 2/3 | ___ 2 | 2/3 ___ | \/ 3 2*\/ 2 *\/ 3 *\/ x | | 2 *log|\/ x + ----| 2/3 / 2/3 5/6 ___\ 2 *\/ 3 *atan|- ----- + -------------------| | 1 \ 2 / 2 *log\8*2 + 16*x - 8*2 *\/ x / \ 3 3 / | ----------------- dx = C - ---------------------- + -------------------------------------- + ---------------------------------------------- | ___ ___ 3 6 3 | x*\/ 2 *\/ x + 1 | /
$$\int \frac{1}{x \sqrt{2} \sqrt{x} + 1}\, dx = C - \frac{2^{\frac{2}{3}} \log{\left(\sqrt{x} + \frac{2^{\frac{5}{6}}}{2} \right)}}{3} + \frac{2^{\frac{2}{3}} \log{\left(- 8 \cdot 2^{\frac{5}{6}} \sqrt{x} + 16 x + 8 \cdot 2^{\frac{2}{3}} \right)}}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt[6]{2} \sqrt{3} \sqrt{x}}{3} - \frac{\sqrt{3}}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ 5/6\ / 5/6\ / ___ 6 ___ ___\
2/3 | 2 | 2/3 |2 | 2/3 ___ |\/ 3 2*\/ 2 *\/ 3 |
2 *log|1 + ----| 2/3 / 2/3\ 2 *log|----| 2/3 / 5/6 2/3\ 2 *\/ 3 *atan|----- - -------------| 2/3 ___
\ 2 / 2 *log\8*2 / \ 2 / 2 *log\16 - 8*2 + 8*2 / \ 3 3 / pi*2 *\/ 3
- ------------------ - ---------------- + -------------- + ------------------------------ - -------------------------------------- + -------------
3 6 3 6 3 18
$$- \frac{2^{\frac{2}{3}} \log{\left(8 \cdot 2^{\frac{2}{3}} \right)}}{6} - \frac{2^{\frac{2}{3}} \log{\left(\frac{2^{\frac{5}{6}}}{2} + 1 \right)}}{3} + \frac{2^{\frac{2}{3}} \log{\left(\frac{2^{\frac{5}{6}}}{2} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \pi}{18} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(- \frac{2 \sqrt[6]{2} \sqrt{3}}{3} + \frac{\sqrt{3}}{3} \right)}}{3} + \frac{2^{\frac{2}{3}} \log{\left(- 8 \cdot 2^{\frac{5}{6}} + 8 \cdot 2^{\frac{2}{3}} + 16 \right)}}{6}$$
=
=
/ 5/6\ / 5/6\ / ___ 6 ___ ___\
2/3 | 2 | 2/3 |2 | 2/3 ___ |\/ 3 2*\/ 2 *\/ 3 |
2 *log|1 + ----| 2/3 / 2/3\ 2 *log|----| 2/3 / 5/6 2/3\ 2 *\/ 3 *atan|----- - -------------| 2/3 ___
\ 2 / 2 *log\8*2 / \ 2 / 2 *log\16 - 8*2 + 8*2 / \ 3 3 / pi*2 *\/ 3
- ------------------ - ---------------- + -------------- + ------------------------------ - -------------------------------------- + -------------
3 6 3 6 3 18
$$- \frac{2^{\frac{2}{3}} \log{\left(8 \cdot 2^{\frac{2}{3}} \right)}}{6} - \frac{2^{\frac{2}{3}} \log{\left(\frac{2^{\frac{5}{6}}}{2} + 1 \right)}}{3} + \frac{2^{\frac{2}{3}} \log{\left(\frac{2^{\frac{5}{6}}}{2} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \pi}{18} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(- \frac{2 \sqrt[6]{2} \sqrt{3}}{3} + \frac{\sqrt{3}}{3} \right)}}{3} + \frac{2^{\frac{2}{3}} \log{\left(- 8 \cdot 2^{\frac{5}{6}} + 8 \cdot 2^{\frac{2}{3}} + 16 \right)}}{6}$$
-2^(2/3)*log(1 + 2^(5/6)/2)/3 - 2^(2/3)*log(8*2^(2/3))/6 + 2^(2/3)*log(2^(5/6)/2)/3 + 2^(2/3)*log(16 - 8*2^(5/6) + 8*2^(2/3))/6 - 2^(2/3)*sqrt(3)*atan(sqrt(3)/3 - 2*2^(1/6)*sqrt(3)/3)/3 + pi*2^(2/3)*sqrt(3)/18
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.