Sr Examen
Integral de 1/(x^3+9) dx
Solución
Ha introducido
[src]
7 / | | 1 | ------ dx | 3 | x + 9 | / 3
$$\int\limits_{3}^{7} \frac{1}{x^{3} + 9}\, dx$$
Integral(1/(x^3 + 9), (x, 3, 7))
Respuesta (Indefinida)
[src]
/ ___ 5/6\ / 6 ___ | \/ 3 2*x*3 | | 2/3 / 2 3 ___ 2/3\ \/ 3 *atan|- ----- + --------| 2/3 / 2/3\ | 1 3 *log\x + 3*\/ 3 - x*3 / \ 3 9 / 3 *log\x + 3 / | ------ dx = C - ------------------------------- + ------------------------------ + ------------------ | 3 54 9 27 | x + 9 | /
$$\int \frac{1}{x^{3} + 9}\, dx = C + \frac{3^{\frac{2}{3}} \log{\left(x + 3^{\frac{2}{3}} \right)}}{27} - \frac{3^{\frac{2}{3}} \log{\left(x^{2} - 3^{\frac{2}{3}} x + 3 \sqrt[3]{3} \right)}}{54} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \cdot 3^{\frac{5}{6}} x}{9} - \frac{\sqrt{3}}{3} \right)}}{9}$$
Gráfica
Respuesta
[src]
/ 5/6 ___\ / 5/6 ___\
6 ___ | 14*3 \/ 3 | 6 ___ | 2*3 \/ 3 |
\/ 3 *atan|- ------- + -----| 2/3 / 2/3\ 2/3 / 2/3 3 ___\ \/ 3 *atan|- ------ + -----| 2/3 / 2/3\ 2/3 / 2/3 3 ___\
\ 9 3 / 3 *log\3 + 3 / 3 *log\49 - 7*3 + 3*\/ 3 / \ 3 3 / 3 *log\7 + 3 / 3 *log\9 - 3*3 + 3*\/ 3 /
- ----------------------------- - ------------------ - ------------------------------- + ---------------------------- + ------------------ + ------------------------------
9 27 54 9 27 54
$$- \frac{3^{\frac{2}{3}} \log{\left(- 7 \cdot 3^{\frac{2}{3}} + 3 \sqrt[3]{3} + 49 \right)}}{54} - \frac{3^{\frac{2}{3}} \log{\left(3^{\frac{2}{3}} + 3 \right)}}{27} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(- \frac{2 \cdot 3^{\frac{5}{6}}}{3} + \frac{\sqrt{3}}{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(- 3 \cdot 3^{\frac{2}{3}} + 3 \sqrt[3]{3} + 9 \right)}}{54} + \frac{3^{\frac{2}{3}} \log{\left(3^{\frac{2}{3}} + 7 \right)}}{27} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(- \frac{14 \cdot 3^{\frac{5}{6}}}{9} + \frac{\sqrt{3}}{3} \right)}}{9}$$
=
=
/ 5/6 ___\ / 5/6 ___\
6 ___ | 14*3 \/ 3 | 6 ___ | 2*3 \/ 3 |
\/ 3 *atan|- ------- + -----| 2/3 / 2/3\ 2/3 / 2/3 3 ___\ \/ 3 *atan|- ------ + -----| 2/3 / 2/3\ 2/3 / 2/3 3 ___\
\ 9 3 / 3 *log\3 + 3 / 3 *log\49 - 7*3 + 3*\/ 3 / \ 3 3 / 3 *log\7 + 3 / 3 *log\9 - 3*3 + 3*\/ 3 /
- ----------------------------- - ------------------ - ------------------------------- + ---------------------------- + ------------------ + ------------------------------
9 27 54 9 27 54
$$- \frac{3^{\frac{2}{3}} \log{\left(- 7 \cdot 3^{\frac{2}{3}} + 3 \sqrt[3]{3} + 49 \right)}}{54} - \frac{3^{\frac{2}{3}} \log{\left(3^{\frac{2}{3}} + 3 \right)}}{27} + \frac{\sqrt[6]{3} \operatorname{atan}{\left(- \frac{2 \cdot 3^{\frac{5}{6}}}{3} + \frac{\sqrt{3}}{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(- 3 \cdot 3^{\frac{2}{3}} + 3 \sqrt[3]{3} + 9 \right)}}{54} + \frac{3^{\frac{2}{3}} \log{\left(3^{\frac{2}{3}} + 7 \right)}}{27} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(- \frac{14 \cdot 3^{\frac{5}{6}}}{9} + \frac{\sqrt{3}}{3} \right)}}{9}$$
-3^(1/6)*atan(-14*3^(5/6)/9 + sqrt(3)/3)/9 - 3^(2/3)*log(3 + 3^(2/3))/27 - 3^(2/3)*log(49 - 7*3^(2/3) + 3*3^(1/3))/54 + 3^(1/6)*atan(-2*3^(5/6)/3 + sqrt(3)/3)/9 + 3^(2/3)*log(7 + 3^(2/3))/27 + 3^(2/3)*log(9 - 3*3^(2/3) + 3*3^(1/3))/54
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.