Sr Examen
Integral de 1/(3*x^3+1) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 3 | 3*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{1}{3 x^{3} + 1}\, dx$$
Integral(1/(3*x^3 + 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ 3 ___ 2/3\ / ___ 5/6\ / 2/3\ / 2/3 | 2 \/ 3 x*3 | 6 ___ | \/ 3 2*x*3 | 2/3 | 3 | | 3 *log|x + ----- - ------| \/ 3 *atan|- ----- + --------| 3 *log|x + ----| | 1 \ 3 3 / \ 3 3 / \ 3 / | -------- dx = C - ----------------------------- + ------------------------------ + ------------------ | 3 18 3 9 | 3*x + 1 | /
$$\int \frac{1}{3 x^{3} + 1}\, dx = C + \frac{3^{\frac{2}{3}} \log{\left(x + \frac{3^{\frac{2}{3}}}{3} \right)}}{9} - \frac{3^{\frac{2}{3}} \log{\left(x^{2} - \frac{3^{\frac{2}{3}} x}{3} + \frac{\sqrt[3]{3}}{3} \right)}}{18} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \cdot 3^{\frac{5}{6}} x}{3} - \frac{\sqrt{3}}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ 5/6 ___\ / 2/3\ / 2/3 3 ___\ / 2/3\ /3 ___\
6 ___ | 2*3 \/ 3 | 2/3 |3 | 2/3 | 3 \/ 3 | 2/3 | 3 | 2/3 |\/ 3 |
\/ 3 *atan|- ------ + -----| 3 *log|----| 3 *log|1 - ---- + -----| 3 *log|1 + ----| 6 ___ 3 *log|-----|
\ 3 3 / \ 3 / \ 3 3 / \ 3 / pi*\/ 3 \ 3 /
- ---------------------------- - -------------- - -------------------------- + ------------------ + -------- + ---------------
3 9 18 9 18 18
$$\frac{3^{\frac{2}{3}} \log{\left(\frac{\sqrt[3]{3}}{3} \right)}}{18} - \frac{3^{\frac{2}{3}} \log{\left(- \frac{3^{\frac{2}{3}}}{3} + \frac{\sqrt[3]{3}}{3} + 1 \right)}}{18} - \frac{3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{2}{3}}}{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{2}{3}}}{3} + 1 \right)}}{9} + \frac{\sqrt[6]{3} \pi}{18} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(- \frac{2 \cdot 3^{\frac{5}{6}}}{3} + \frac{\sqrt{3}}{3} \right)}}{3}$$
=
=
/ 5/6 ___\ / 2/3\ / 2/3 3 ___\ / 2/3\ /3 ___\
6 ___ | 2*3 \/ 3 | 2/3 |3 | 2/3 | 3 \/ 3 | 2/3 | 3 | 2/3 |\/ 3 |
\/ 3 *atan|- ------ + -----| 3 *log|----| 3 *log|1 - ---- + -----| 3 *log|1 + ----| 6 ___ 3 *log|-----|
\ 3 3 / \ 3 / \ 3 3 / \ 3 / pi*\/ 3 \ 3 /
- ---------------------------- - -------------- - -------------------------- + ------------------ + -------- + ---------------
3 9 18 9 18 18
$$\frac{3^{\frac{2}{3}} \log{\left(\frac{\sqrt[3]{3}}{3} \right)}}{18} - \frac{3^{\frac{2}{3}} \log{\left(- \frac{3^{\frac{2}{3}}}{3} + \frac{\sqrt[3]{3}}{3} + 1 \right)}}{18} - \frac{3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{2}{3}}}{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{2}{3}}}{3} + 1 \right)}}{9} + \frac{\sqrt[6]{3} \pi}{18} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(- \frac{2 \cdot 3^{\frac{5}{6}}}{3} + \frac{\sqrt{3}}{3} \right)}}{3}$$
-3^(1/6)*atan(-2*3^(5/6)/3 + sqrt(3)/3)/3 - 3^(2/3)*log(3^(2/3)/3)/9 - 3^(2/3)*log(1 - 3^(2/3)/3 + 3^(1/3)/3)/18 + 3^(2/3)*log(1 + 3^(2/3)/3)/9 + pi*3^(1/6)/18 + 3^(2/3)*log(3^(1/3)/3)/18
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.