Sr Examen
Integral de cbrt(x)/(3*x+1) dx
Solución
Ha introducido
[src]
o / | | 3 ___ | \/ x | ------- dx | 3*x + 1 | / -1
$$\int\limits_{-1}^{o} \frac{\sqrt[3]{x}}{3 x + 1}\, dx$$
Integral(x^(1/3)/(3*x + 1), (x, -1, o))
Respuesta (Indefinida)
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/ / ___ 5/6 3 ___\ / 2/3\ | 6 ___ | \/ 3 2*3 *\/ x | 2/3 |3 ___ 3 | | 3 ___ \/ 3 *atan|- ----- + ------------| 3 *log|\/ x + ----| 2/3 / 3 ___ 2/3 2/3 3 ___\ | \/ x 3 ___ \ 3 3 / \ 3 / 3 *log\12*\/ 3 + 36*x - 12*3 *\/ x / | ------- dx = C + \/ x - ---------------------------------- - ---------------------- + -------------------------------------------- | 3*x + 1 3 9 18 | /
$$\int \frac{\sqrt[3]{x}}{3 x + 1}\, dx = C + \sqrt[3]{x} - \frac{3^{\frac{2}{3}} \log{\left(\sqrt[3]{x} + \frac{3^{\frac{2}{3}}}{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(36 x^{\frac{2}{3}} - 12 \cdot 3^{\frac{2}{3}} \sqrt[3]{x} + 12 \sqrt[3]{3} \right)}}{18} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \cdot 3^{\frac{5}{6}} \sqrt[3]{x}}{3} - \frac{\sqrt{3}}{3} \right)}}{3}$$
Respuesta
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/ ___ 5/6 3 ___\ / ___ 3 ____ 5/6\ / 2/3\ / 2/3\
6 ___ | \/ 3 2*3 *\/ o | 6 ___ |\/ 3 2*\/ -1 *3 | 2/3 |3 ___ 3 | 2/3 |3 ____ 3 |
\/ 3 *atan|- ----- + ------------| \/ 3 *atan|----- - -------------| 3 *log|\/ o + ----| 2/3 / 3 ___ 2/3 3 ____ 2/3\ 3 *log|\/ -1 + ----| 2/3 / 3 ___ 2/3 2/3 3 ___\
3 ___ 3 ____ \ 3 3 / \ 3 3 / \ 3 / 3 *log\12*\/ 3 + 36*(-1) - 12*\/ -1 *3 / \ 3 / 3 *log\12*\/ 3 + 36*o - 12*3 *\/ o /
\/ o - \/ -1 - ---------------------------------- - --------------------------------- - ---------------------- - ------------------------------------------------ + ----------------------- + --------------------------------------------
3 3 9 18 9 18
$$\sqrt[3]{o} - \frac{3^{\frac{2}{3}} \log{\left(\sqrt[3]{o} + \frac{3^{\frac{2}{3}}}{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(36 o^{\frac{2}{3}} - 12 \cdot 3^{\frac{2}{3}} \sqrt[3]{o} + 12 \sqrt[3]{3} \right)}}{18} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \cdot 3^{\frac{5}{6}} \sqrt[3]{o}}{3} - \frac{\sqrt{3}}{3} \right)}}{3} - \sqrt[3]{-1} - \frac{3^{\frac{2}{3}} \log{\left(12 \sqrt[3]{3} - 12 \sqrt[3]{-1} \cdot 3^{\frac{2}{3}} + 36 \left(-1\right)^{\frac{2}{3}} \right)}}{18} + \frac{3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{2}{3}}}{3} + \sqrt[3]{-1} \right)}}{9} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} - \frac{2 \sqrt[3]{-1} \cdot 3^{\frac{5}{6}}}{3} \right)}}{3}$$
=
=
/ ___ 5/6 3 ___\ / ___ 3 ____ 5/6\ / 2/3\ / 2/3\
6 ___ | \/ 3 2*3 *\/ o | 6 ___ |\/ 3 2*\/ -1 *3 | 2/3 |3 ___ 3 | 2/3 |3 ____ 3 |
\/ 3 *atan|- ----- + ------------| \/ 3 *atan|----- - -------------| 3 *log|\/ o + ----| 2/3 / 3 ___ 2/3 3 ____ 2/3\ 3 *log|\/ -1 + ----| 2/3 / 3 ___ 2/3 2/3 3 ___\
3 ___ 3 ____ \ 3 3 / \ 3 3 / \ 3 / 3 *log\12*\/ 3 + 36*(-1) - 12*\/ -1 *3 / \ 3 / 3 *log\12*\/ 3 + 36*o - 12*3 *\/ o /
\/ o - \/ -1 - ---------------------------------- - --------------------------------- - ---------------------- - ------------------------------------------------ + ----------------------- + --------------------------------------------
3 3 9 18 9 18
$$\sqrt[3]{o} - \frac{3^{\frac{2}{3}} \log{\left(\sqrt[3]{o} + \frac{3^{\frac{2}{3}}}{3} \right)}}{9} + \frac{3^{\frac{2}{3}} \log{\left(36 o^{\frac{2}{3}} - 12 \cdot 3^{\frac{2}{3}} \sqrt[3]{o} + 12 \sqrt[3]{3} \right)}}{18} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{2 \cdot 3^{\frac{5}{6}} \sqrt[3]{o}}{3} - \frac{\sqrt{3}}{3} \right)}}{3} - \sqrt[3]{-1} - \frac{3^{\frac{2}{3}} \log{\left(12 \sqrt[3]{3} - 12 \sqrt[3]{-1} \cdot 3^{\frac{2}{3}} + 36 \left(-1\right)^{\frac{2}{3}} \right)}}{18} + \frac{3^{\frac{2}{3}} \log{\left(\frac{3^{\frac{2}{3}}}{3} + \sqrt[3]{-1} \right)}}{9} - \frac{\sqrt[6]{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} - \frac{2 \sqrt[3]{-1} \cdot 3^{\frac{5}{6}}}{3} \right)}}{3}$$
o^(1/3) - (-1)^(1/3) - 3^(1/6)*atan(-sqrt(3)/3 + 2*3^(5/6)*o^(1/3)/3)/3 - 3^(1/6)*atan(sqrt(3)/3 - 2*(-1)^(1/3)*3^(5/6)/3)/3 - 3^(2/3)*log(o^(1/3) + 3^(2/3)/3)/9 - 3^(2/3)*log(12*3^(1/3) + 36*(-1)^(2/3) - 12*(-1)^(1/3)*3^(2/3))/18 + 3^(2/3)*log((-1)^(1/3) + 3^(2/3)/3)/9 + 3^(2/3)*log(12*3^(1/3) + 36*o^(2/3) - 12*3^(2/3)*o^(1/3))/18
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.