Sr Examen
Integral de 1/(2x^3+3) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 3 | 2*x + 3 | / 0
$$\int\limits_{0}^{1} \frac{1}{2 x^{3} + 3}\, dx$$
Integral(1/(2*x^3 + 3), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ 3 ____\ / ___ 3 ___ 6 ___\ / 3 ____ | \/ 12 | 2/3 5/6 | \/ 3 2*x*\/ 2 *\/ 3 | | / 3 ____ 2/3 3 ___\ / 3 ____ 3 ____\ \/ 12 *log|x + ------| 2 *3 *atan|- ----- + ---------------| | 1 | \/ 12 2 *\/ 3 | | 2 \/ 18 x*\/ 12 | \ 2 / \ 3 3 / | -------- dx = C + |- ------ - ----------|*log|x + ------ - --------| + ---------------------- + ----------------------------------------- | 3 \ 144 48 / \ 2 2 / 18 18 | 2*x + 3 | /
$$\int \frac{1}{2 x^{3} + 3}\, dx = C + \frac{\sqrt[3]{12} \log{\left(x + \frac{\sqrt[3]{12}}{2} \right)}}{18} + \left(- \frac{2^{\frac{2}{3}} \sqrt[3]{3}}{48} - \frac{\sqrt[3]{12}}{144}\right) \log{\left(x^{2} - \frac{\sqrt[3]{12} x}{2} + \frac{\sqrt[3]{18}}{2} \right)} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2 \sqrt[3]{2} \sqrt[6]{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{18}$$
Gráfica
Respuesta
[src]
/3 ____\ / 3 ____\ / ___ 3 ___ 6 ___\
3 ____ |\/ 12 | 3 ____ | \/ 12 | 2/3 5/6 |\/ 3 2*\/ 2 *\/ 3 |
/ 3 ____ 2/3 3 ___\ / 3 ____ 3 ____\ / 3 ____ 2/3 3 ___\ /3 ____\ \/ 12 *log|------| \/ 12 *log|1 + ------| 2 *3 *atan|----- - -------------| 2/3 5/6
| \/ 12 2 *\/ 3 | | \/ 18 \/ 12 | | \/ 12 2 *\/ 3 | |\/ 18 | \ 2 / \ 2 / \ 3 3 / pi*2 *3
|- ------ - ----------|*log|1 + ------ - ------| - |- ------ - ----------|*log|------| - ------------------ + ---------------------- - ------------------------------------- + ------------
\ 144 48 / \ 2 2 / \ 144 48 / \ 2 / 18 18 18 108
$$- \frac{\sqrt[3]{12} \log{\left(\frac{\sqrt[3]{12}}{2} \right)}}{18} + \left(- \frac{2^{\frac{2}{3}} \sqrt[3]{3}}{48} - \frac{\sqrt[3]{12}}{144}\right) \log{\left(- \frac{\sqrt[3]{12}}{2} + 1 + \frac{\sqrt[3]{18}}{2} \right)} - \left(- \frac{2^{\frac{2}{3}} \sqrt[3]{3}}{48} - \frac{\sqrt[3]{12}}{144}\right) \log{\left(\frac{\sqrt[3]{18}}{2} \right)} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \operatorname{atan}{\left(- \frac{2 \sqrt[3]{2} \sqrt[6]{3}}{3} + \frac{\sqrt{3}}{3} \right)}}{18} + \frac{\sqrt[3]{12} \log{\left(1 + \frac{\sqrt[3]{12}}{2} \right)}}{18} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \pi}{108}$$
=
=
/3 ____\ / 3 ____\ / ___ 3 ___ 6 ___\
3 ____ |\/ 12 | 3 ____ | \/ 12 | 2/3 5/6 |\/ 3 2*\/ 2 *\/ 3 |
/ 3 ____ 2/3 3 ___\ / 3 ____ 3 ____\ / 3 ____ 2/3 3 ___\ /3 ____\ \/ 12 *log|------| \/ 12 *log|1 + ------| 2 *3 *atan|----- - -------------| 2/3 5/6
| \/ 12 2 *\/ 3 | | \/ 18 \/ 12 | | \/ 12 2 *\/ 3 | |\/ 18 | \ 2 / \ 2 / \ 3 3 / pi*2 *3
|- ------ - ----------|*log|1 + ------ - ------| - |- ------ - ----------|*log|------| - ------------------ + ---------------------- - ------------------------------------- + ------------
\ 144 48 / \ 2 2 / \ 144 48 / \ 2 / 18 18 18 108
$$- \frac{\sqrt[3]{12} \log{\left(\frac{\sqrt[3]{12}}{2} \right)}}{18} + \left(- \frac{2^{\frac{2}{3}} \sqrt[3]{3}}{48} - \frac{\sqrt[3]{12}}{144}\right) \log{\left(- \frac{\sqrt[3]{12}}{2} + 1 + \frac{\sqrt[3]{18}}{2} \right)} - \left(- \frac{2^{\frac{2}{3}} \sqrt[3]{3}}{48} - \frac{\sqrt[3]{12}}{144}\right) \log{\left(\frac{\sqrt[3]{18}}{2} \right)} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \operatorname{atan}{\left(- \frac{2 \sqrt[3]{2} \sqrt[6]{3}}{3} + \frac{\sqrt{3}}{3} \right)}}{18} + \frac{\sqrt[3]{12} \log{\left(1 + \frac{\sqrt[3]{12}}{2} \right)}}{18} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \pi}{108}$$
(-12^(1/3)/144 - 2^(2/3)*3^(1/3)/48)*log(1 + 18^(1/3)/2 - 12^(1/3)/2) - (-12^(1/3)/144 - 2^(2/3)*3^(1/3)/48)*log(18^(1/3)/2) - 12^(1/3)*log(12^(1/3)/2)/18 + 12^(1/3)*log(1 + 12^(1/3)/2)/18 - 2^(2/3)*3^(5/6)*atan(sqrt(3)/3 - 2*2^(1/3)*3^(1/6)/3)/18 + pi*2^(2/3)*3^(5/6)/108
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.