Sr Examen
Integral de 1/(((x^2+3x)^2)+(2x+3x^2)) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------------------ dx | 2 | / 2 \ 2 | \x + 3*x/ + 2*x + 3*x | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 3 x\right)^{2} + \left(3 x^{2} + 2 x\right)}\, dx$$
Integral(1/((x^2 + 3*x)^2 + 2*x + 3*x^2), (x, 0, 1))
Respuesta (Indefinida)
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/ 2 \ / ____________ ____________ ____________ \
| / 2/3 3 ___\ | _______________________ | / 3 ___ 3 ___ / 3 ___ / 3 ___ |
| |1 2*6 2*\/ 6 | | / 3 ___ 2/3\ / 2/3 3 ___ \ ___ / 2/3 3 ___ | 28*\/ -1 + \/ 6 7*\/ 6 *\/ -1 + \/ 6 14*x*\/ -1 + \/ 6 |
/ 2/3 3 ___\ | 81*|- - ------ + -------| 2/3 3 ___| |1 \/ 6 6 | | 2 2 / 3 ___\ 6 4*\/ 6 | / ___ 2/3 6 ___ 2/3 6 ___\ \/ 3 *\/ -6 + 5*6 + 6*\/ 6 *atan|------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------|
/ |1 2*6 2*\/ 6 | |17 \3 9 9 / 2*6 2*\/ 6 | |- - ----- + ----|*log|x - --------------------- + x*\4 + \/ 6 / + --------------------- - ---------------------| 3 ___ 5/6 |\/ 3 2*2 *\/ 3 x*2 *\/ 3 | | _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________|
| / / 2 \\ |- - ------ + -------|*log|-- + x - -------------------------- - ------ + -------| \3 9 9 / | 3 ___ 2/3 3 ___ 2/3 3 ___ 2/3| 3 ___ / 3 ___\ 3 ___ / 2/3 2 3 ___ / 3 ___\\ \/ 2 *3 *atan|----- + ------------ + ------------| | ___ / 2/3 3 ___ 3 ___ 5/6 / 2/3 3 ___ 2/3 6 ___ / 2/3 3 ___ ___ / 2/3 3 ___ 3 ___ 5/6 / 2/3 3 ___ 2/3 6 ___ / 2/3 3 ___ ___ / 2/3 3 ___ 3 ___ 5/6 / 2/3 3 ___ 2/3 6 ___ / 2/3 3 ___ |
| 1 log(x) | 3 |2 162*t 36*t|| \3 9 9 / \28 28 7 7 / \ -5 - 3*\/ 6 + 3*6 -5 - 3*\/ 6 + 3*6 -5 - 3*\/ 6 + 3*6 / \/ 6 *log\2 + x - \/ 6 / \/ 6 *log\4 + 6 + x + 2*\/ 6 + x*\4 + \/ 6 // \ 3 3 3 / \\/ 3 *\/ -6 + 5*6 + 6*\/ 6 - \/ 2 *3 *\/ -6 + 5*6 + 6*\/ 6 + 3*2 *\/ 3 *\/ -6 + 5*6 + 6*\/ 6 \/ 3 *\/ -6 + 5*6 + 6*\/ 6 - \/ 2 *3 *\/ -6 + 5*6 + 6*\/ 6 + 3*2 *\/ 3 *\/ -6 + 5*6 + 6*\/ 6 \/ 3 *\/ -6 + 5*6 + 6*\/ 6 - \/ 2 *3 *\/ -6 + 5*6 + 6*\/ 6 + 3*2 *\/ 3 *\/ -6 + 5*6 + 6*\/ 6 /
| ------------------------ dx = C + ------ - 3*RootSum|486*t + 54*t + 1, t -> t*log|- + x - ------ + ----|| - ---------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------ - ------------------------ + -------------------------------------------------- + ---------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| 2 2 \ \7 7 7 // 2 2 3 6 3 ____________
| / 2 \ 2 / 3 ___
| \x + 3*x/ + 2*x + 3*x 9*\/ -1 + \/ 6
|
/
$$\int \frac{1}{\left(x^{2} + 3 x\right)^{2} + \left(3 x^{2} + 2 x\right)}\, dx = C + \frac{\log{\left(x \right)}}{2} - \frac{\sqrt[3]{6} \log{\left(x - \sqrt[3]{6} + 2 \right)}}{3} - \frac{\left(- \frac{2 \cdot 6^{\frac{2}{3}}}{9} + \frac{1}{3} + \frac{2 \sqrt[3]{6}}{9}\right) \log{\left(x - \frac{2 \cdot 6^{\frac{2}{3}}}{7} - \frac{81 \left(- \frac{2 \cdot 6^{\frac{2}{3}}}{9} + \frac{1}{3} + \frac{2 \sqrt[3]{6}}{9}\right)^{2}}{28} + \frac{2 \sqrt[3]{6}}{7} + \frac{17}{28} \right)}}{2} + \frac{\sqrt[3]{6} \log{\left(x^{2} + x \left(\sqrt[3]{6} + 4\right) + 6^{\frac{2}{3}} + 2 \sqrt[3]{6} + 4 \right)}}{6} - \frac{\left(- \frac{\sqrt[3]{6}}{9} + \frac{1}{3} + \frac{6^{\frac{2}{3}}}{9}\right) \log{\left(x^{2} + x \left(\sqrt[3]{6} + 4\right) + \frac{6^{\frac{2}{3}}}{- 3 \sqrt[3]{6} - 5 + 3 \cdot 6^{\frac{2}{3}}} - \frac{2}{- 3 \sqrt[3]{6} - 5 + 3 \cdot 6^{\frac{2}{3}}} - \frac{4 \sqrt[3]{6}}{- 3 \sqrt[3]{6} - 5 + 3 \cdot 6^{\frac{2}{3}}} \right)}}{2} + \frac{\sqrt[3]{2} \cdot 3^{\frac{5}{6}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt[6]{3} x}{3} + \frac{\sqrt{3}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \sqrt[6]{3}}{3} \right)}}{3} + \frac{\sqrt{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} \operatorname{atan}{\left(\frac{14 x \sqrt{-1 + \sqrt[3]{6}}}{- \sqrt[3]{2} \cdot 3^{\frac{5}{6}} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + \sqrt{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + 3 \cdot 2^{\frac{2}{3}} \sqrt[6]{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}}} + \frac{7 \sqrt[3]{6} \sqrt{-1 + \sqrt[3]{6}}}{- \sqrt[3]{2} \cdot 3^{\frac{5}{6}} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + \sqrt{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + 3 \cdot 2^{\frac{2}{3}} \sqrt[6]{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}}} + \frac{28 \sqrt{-1 + \sqrt[3]{6}}}{- \sqrt[3]{2} \cdot 3^{\frac{5}{6}} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + \sqrt{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}} + 3 \cdot 2^{\frac{2}{3}} \sqrt[6]{3} \sqrt{-6 + 6 \sqrt[3]{6} + 5 \cdot 6^{\frac{2}{3}}}} \right)}}{9 \sqrt{-1 + \sqrt[3]{6}}} - 3 \operatorname{RootSum} {\left(486 t^{3} + 54 t + 1, \left( t \mapsto t \log{\left(- \frac{162 t^{2}}{7} + \frac{36 t}{7} + x + \frac{2}{7} \right)} \right)\right)}$$
Gráfica
Respuesta
[src]
1 / | | 1 | ------------------------ dx | 2 | / 2 \ 2 | \x + 3*x/ + 2*x + 3*x | / 0
$$\int\limits_{0}^{1} \frac{1}{3 x^{2} + 2 x + \left(x^{2} + 3 x\right)^{2}}\, dx$$
=
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1 / | | 1 | ------------------------ dx | 2 | / 2 \ 2 | \x + 3*x/ + 2*x + 3*x | / 0
$$\int\limits_{0}^{1} \frac{1}{3 x^{2} + 2 x + \left(x^{2} + 3 x\right)^{2}}\, dx$$
Integral(1/((x^2 + 3*x)^2 + 2*x + 3*x^2), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.