Sr Examen
Integral de 1/(x*(sqrt(3)*sqrt(x))+1) dx
Solución
Ha introducido
[src]
1 / | | 1 | ----------------- dx | ___ ___ | x*\/ 3 *\/ x + 1 | / 0
$$\int\limits_{0}^{1} \frac{1}{x \sqrt{3} \sqrt{x} + 1}\, dx$$
Integral(1/(x*(sqrt(3)*sqrt(x)) + 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ 5/6\ / ___ 2/3 ___\ / 2/3 | ___ 3 | 6 ___ | \/ 3 2*3 *\/ x | | 2*3 *log|\/ x + ----| 2/3 / 2/3 5/6 ___\ 2*\/ 3 *atan|- ----- + ------------| | 1 \ 3 / 3 *log\12*3 + 36*x - 12*3 *\/ x / \ 3 3 / | ----------------- dx = C - ------------------------ + ---------------------------------------- + ------------------------------------ | ___ ___ 9 9 3 | x*\/ 3 *\/ x + 1 | /
$$\int \frac{1}{x \sqrt{3} \sqrt{x} + 1}\, dx = C - \frac{2 \cdot 3^{\frac{2}{3}} \log{\left(\sqrt{x} + \frac{3^{\frac{5}{6}}}{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(- 12 \cdot 3^{\frac{5}{6}} \sqrt{x} + 36 x + 12 \cdot 3^{\frac{2}{3}} \right)}}{9} + \frac{2 \sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \cdot 3^{\frac{2}{3}} \sqrt{x}}{3} - \frac{\sqrt{3}}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ 2/3 ___\ / 5/6\ / 5/6\
6 ___ | 2*3 \/ 3 | 2/3 | 3 | 2/3 |3 |
2*\/ 3 *atan|- ------ + -----| 2*3 *log|1 + ----| 2/3 / 2/3\ 6 ___ 2/3 / 5/6 2/3\ 2*3 *log|----|
\ 3 3 / \ 3 / 3 *log\12*3 / pi*\/ 3 3 *log\36 - 12*3 + 12*3 / \ 3 /
- ------------------------------ - -------------------- - ----------------- + -------- + -------------------------------- + ----------------
3 9 9 9 9 9
$$- \frac{3^{\frac{2}{3}} \log{\left(12 \cdot 3^{\frac{2}{3}} \right)}}{9} - \frac{2 \cdot 3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{5}{6}}}{3} + 1 \right)}}{9} + \frac{2 \cdot 3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{5}{6}}}{3} \right)}}{9} + \frac{\sqrt[6]{3} \pi}{9} - \frac{2 \sqrt[6]{3} \operatorname{atan}{\left(- \frac{2 \cdot 3^{\frac{2}{3}}}{3} + \frac{\sqrt{3}}{3} \right)}}{3} + \frac{3^{\frac{2}{3}} \log{\left(- 12 \cdot 3^{\frac{5}{6}} + 12 \cdot 3^{\frac{2}{3}} + 36 \right)}}{9}$$
=
=
/ 2/3 ___\ / 5/6\ / 5/6\
6 ___ | 2*3 \/ 3 | 2/3 | 3 | 2/3 |3 |
2*\/ 3 *atan|- ------ + -----| 2*3 *log|1 + ----| 2/3 / 2/3\ 6 ___ 2/3 / 5/6 2/3\ 2*3 *log|----|
\ 3 3 / \ 3 / 3 *log\12*3 / pi*\/ 3 3 *log\36 - 12*3 + 12*3 / \ 3 /
- ------------------------------ - -------------------- - ----------------- + -------- + -------------------------------- + ----------------
3 9 9 9 9 9
$$- \frac{3^{\frac{2}{3}} \log{\left(12 \cdot 3^{\frac{2}{3}} \right)}}{9} - \frac{2 \cdot 3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{5}{6}}}{3} + 1 \right)}}{9} + \frac{2 \cdot 3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{5}{6}}}{3} \right)}}{9} + \frac{\sqrt[6]{3} \pi}{9} - \frac{2 \sqrt[6]{3} \operatorname{atan}{\left(- \frac{2 \cdot 3^{\frac{2}{3}}}{3} + \frac{\sqrt{3}}{3} \right)}}{3} + \frac{3^{\frac{2}{3}} \log{\left(- 12 \cdot 3^{\frac{5}{6}} + 12 \cdot 3^{\frac{2}{3}} + 36 \right)}}{9}$$
-2*3^(1/6)*atan(-2*3^(2/3)/3 + sqrt(3)/3)/3 - 2*3^(2/3)*log(1 + 3^(5/6)/3)/9 - 3^(2/3)*log(12*3^(2/3))/9 + pi*3^(1/6)/9 + 3^(2/3)*log(36 - 12*3^(5/6) + 12*3^(2/3))/9 + 2*3^(2/3)*log(3^(5/6)/3)/9
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.