Sr Examen
Integral de (x^2+x+1)/(x^4+2x^2+3) dx
Solución
Ha introducido
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1 / | | 2 | x + x + 1 | ------------- dx | 4 2 | x + 2*x + 3 | / 0
$$\int\limits_{0}^{1} \frac{\left(x^{2} + x\right) + 1}{\left(x^{4} + 2 x^{2}\right) + 3}\, dx$$
Integral((x^2 + x + 1)/(x^4 + 2*x^2 + 3), (x, 0, 1))
Respuesta (Indefinida)
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/ _____________________________________ _____________________________________ ____________________________________ ____________________________________ / ___ / 2\\ | ____________ _____________ / ___________\ ____________ _____________ / ___________\ / ___________ / ___________ ___________ ___________ \ / ___________ / ___________ ___________ ___________ \ / ___________ / ___________ ___________ ___________ \ / ___________ / ___________ ___________ ___________ \ ___ |\/ 2 *\1 + x /| | 2 / ___ / / ___________ ___________ ___________ ___________\ ___________ ___________\ / ___ | ___ / ___________ ___________ ___________\ ___________ ___ / ___ | / ___ / / ___________ ___________ ___________ ___________\ ___________ ___________\ / ___ | ___ / ___________ ___________ ___________\ ___________ ___ / ___ | / ___ ___ / ___ | / ___ ___ ___ / ___ ___ / ___ | / ___ ___ / ___ | / ___ ___ ___ / ___ ___ / ___ | / ___ ___ / ___ | ___ / ___ ___ ___ / ___ / ___ | / ___ ___ / ___ | ___ ___ / ___ ___ / ___ / ___ | \/ 2 *atan|--------------| | x + x + 1 / 1 \/ 3 | 2 ___ | / ___ ___ / ___ ___ / ___ / ___ | ___ / ___ ___ / ___ | / 1 \/ 3 | 2 28*\/ 3 | / ___ ___ / ___ / ___ | ___ / ___ 4*\/ 6 *\/ 2 + \/ 3 | / 1 \/ 3 | 2 ___ | / ___ ___ / ___ ___ / ___ / ___ | ___ / ___ ___ / ___ | / 1 \/ 3 | 2 28*\/ 3 | / ___ ___ / ___ / ___ | ___ / ___ 4*\/ 6 *\/ 2 + \/ 3 | / 1 3*\/ 3 \/ 2 *\/ 2 + \/ 3 | 3*\/ 1 + \/ 3 x*\/ 2 \/ 2 *\/ 1 + \/ 3 2*\/ 6 *\/ 1 + \/ 3 | / 1 3*\/ 3 \/ 2 *\/ 2 + \/ 3 | 3*\/ 1 + \/ 3 x*\/ 2 \/ 2 *\/ 1 + \/ 3 2*\/ 6 *\/ 1 + \/ 3 | / 1 \/ 3 \/ 2 *\/ 2 + \/ 3 | 4*\/ 3 *\/ 1 + \/ 3 2*x*\/ 3 3*\/ 2 *\/ 1 + \/ 3 *\/ 2 + \/ 3 | / 1 \/ 3 \/ 2 *\/ 2 + \/ 3 | 2*x*\/ 3 4*\/ 3 *\/ 1 + \/ 3 3*\/ 2 *\/ 1 + \/ 3 *\/ 2 + \/ 3 | \ 2 / | ------------- dx = C + / -- + ----- *log\53 + x + 34*\/ 3 + x*\- 2*\/ 1 + \/ 3 - 4*\/ 3 *\/ 1 + \/ 3 + 3*\/ 2 *\/ 1 + \/ 3 *\/ 2 + \/ 3 / - 23*\/ 2 *\/ 2 + \/ 3 - 10*\/ 6 *\/ 2 + \/ 3 / + / --- + ----- *log|11 + x + -------- + x*\- 4*\/ 1 + \/ 3 + \/ 6 *\/ 1 + \/ 3 *\/ 2 + \/ 3 / - 7*\/ 2 *\/ 2 + \/ 3 - ----------------------| - / -- + ----- *log\53 + x + 34*\/ 3 + x*\2*\/ 1 + \/ 3 + 4*\/ 3 *\/ 1 + \/ 3 - 3*\/ 2 *\/ 1 + \/ 3 *\/ 2 + \/ 3 / - 23*\/ 2 *\/ 2 + \/ 3 - 10*\/ 6 *\/ 2 + \/ 3 / - / --- + ----- *log|11 + x + -------- + x*\4*\/ 1 + \/ 3 - \/ 6 *\/ 1 + \/ 3 *\/ 2 + \/ 3 / - 7*\/ 2 *\/ 2 + \/ 3 - ----------------------| + 2* / -- + ------- - -------------------- *atan|- ------------------------------------------ + --------------------------------------------------------- + --------------------------------------------------------- + ---------------------------------------------------------| + 2* / -- + ------- - -------------------- *atan|------------------------------------------ + --------------------------------------------------------- - --------------------------------------------------------- - ---------------------------------------------------------| + 2* / --- + ----- - -------------------- *atan|- -------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------| + 2* / --- + ----- - -------------------- *atan|-------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------| + -------------------------- | 4 2 \/ 64 64 \/ 192 192 \ 3 3 / \/ 64 64 \/ 192 192 \ 3 3 / \/ 64 64 32 | ______________________________________ ______________________________________ ______________________________________ ______________________________________| \/ 64 64 32 | ______________________________________ ______________________________________ ______________________________________ ______________________________________| \/ 192 64 96 | ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________| \/ 192 64 96 | ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________| 4 | x + 2*x + 3 | / ___________ ___________ / ___________ ___________ / ___________ ___________ / ___________ | | / ___________ ___________ / ___________ ___________ / ___________ ___________ / ___________ | | / ___________ ___________ / ___________ / ___________ ___________ / ___________ / ___________ ___________ / ___________ | | / ___________ ___________ / ___________ / ___________ ___________ / ___________ / ___________ ___________ / ___________ | | | / ___ ___ / ___ / ___ / ___ ___ / ___ / ___ / ___ ___ / ___ / ___ / ___ ___ / ___ | | / ___ ___ / ___ / ___ / ___ ___ / ___ / ___ / ___ ___ / ___ / ___ / ___ ___ / ___ | | / ___ ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ ___ / ___ / ___ ___ / ___ | | / ___ ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ ___ / ___ / ___ ___ / ___ | / \ \/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 \/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 \/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 \/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 / \\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 \/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 \/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 \/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 / \ 2*\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 + \/ 2 *\/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 2*\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 + \/ 2 *\/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 2*\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 + \/ 2 *\/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 / \2*\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 + \/ 2 *\/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 2*\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 + \/ 2 *\/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 2*\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 + \/ 2 *\/ 2 + \/ 3 *\/ 1 + 3*\/ 3 - 2*\/ 2 *\/ 2 + \/ 3 /
$$\int \frac{\left(x^{2} + x\right) + 1}{\left(x^{4} + 2 x^{2}\right) + 3}\, dx = C - \sqrt{\frac{1}{192} + \frac{\sqrt{3}}{192}} \log{\left(x^{2} + x \left(- \sqrt{6} \sqrt{1 + \sqrt{3}} \sqrt{\sqrt{3} + 2} + 4 \sqrt{1 + \sqrt{3}}\right) - 7 \sqrt{2} \sqrt{\sqrt{3} + 2} - \frac{4 \sqrt{6} \sqrt{\sqrt{3} + 2}}{3} + 11 + \frac{28 \sqrt{3}}{3} \right)} + \sqrt{\frac{1}{192} + \frac{\sqrt{3}}{192}} \log{\left(x^{2} + x \left(- 4 \sqrt{1 + \sqrt{3}} + \sqrt{6} \sqrt{1 + \sqrt{3}} \sqrt{\sqrt{3} + 2}\right) - 7 \sqrt{2} \sqrt{\sqrt{3} + 2} - \frac{4 \sqrt{6} \sqrt{\sqrt{3} + 2}}{3} + 11 + \frac{28 \sqrt{3}}{3} \right)} + \sqrt{\frac{1}{64} + \frac{\sqrt{3}}{64}} \log{\left(x^{2} + x \left(- 4 \sqrt{3} \sqrt{1 + \sqrt{3}} - 2 \sqrt{1 + \sqrt{3}} + 3 \sqrt{2} \sqrt{1 + \sqrt{3}} \sqrt{\sqrt{3} + 2}\right) - 23 \sqrt{2} \sqrt{\sqrt{3} + 2} - 10 \sqrt{6} \sqrt{\sqrt{3} + 2} + 53 + 34 \sqrt{3} \right)} - \sqrt{\frac{1}{64} + \frac{\sqrt{3}}{64}} \log{\left(x^{2} + x \left(- 3 \sqrt{2} \sqrt{1 + \sqrt{3}} \sqrt{\sqrt{3} + 2} + 2 \sqrt{1 + \sqrt{3}} + 4 \sqrt{3} \sqrt{1 + \sqrt{3}}\right) - 23 \sqrt{2} \sqrt{\sqrt{3} + 2} - 10 \sqrt{6} \sqrt{\sqrt{3} + 2} + 53 + 34 \sqrt{3} \right)} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x^{2} + 1\right)}{2} \right)}}{4} + 2 \sqrt{- \frac{\sqrt{2} \sqrt{\sqrt{3} + 2}}{96} + \frac{1}{192} + \frac{\sqrt{3}}{64}} \operatorname{atan}{\left(\frac{2 \sqrt{3} x}{2 \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}} + \sqrt{2} \sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} - \frac{4 \sqrt{3} \sqrt{1 + \sqrt{3}}}{2 \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}} + \sqrt{2} \sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} + \frac{3 \sqrt{2} \sqrt{1 + \sqrt{3}} \sqrt{\sqrt{3} + 2}}{2 \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}} + \sqrt{2} \sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} \right)} + 2 \sqrt{- \frac{\sqrt{2} \sqrt{\sqrt{3} + 2}}{96} + \frac{1}{192} + \frac{\sqrt{3}}{64}} \operatorname{atan}{\left(\frac{2 \sqrt{3} x}{2 \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}} + \sqrt{2} \sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} - \frac{3 \sqrt{2} \sqrt{1 + \sqrt{3}} \sqrt{\sqrt{3} + 2}}{2 \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}} + \sqrt{2} \sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} + \frac{4 \sqrt{3} \sqrt{1 + \sqrt{3}}}{2 \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}} + \sqrt{2} \sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} \right)} + 2 \sqrt{- \frac{\sqrt{2} \sqrt{\sqrt{3} + 2}}{32} + \frac{1}{64} + \frac{3 \sqrt{3}}{64}} \operatorname{atan}{\left(\frac{\sqrt{2} x}{\sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} - \frac{3 \sqrt{1 + \sqrt{3}}}{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} + \frac{\sqrt{2} \sqrt{1 + \sqrt{3}}}{\sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} + \frac{2 \sqrt{6} \sqrt{1 + \sqrt{3}}}{\sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} \right)} + 2 \sqrt{- \frac{\sqrt{2} \sqrt{\sqrt{3} + 2}}{32} + \frac{1}{64} + \frac{3 \sqrt{3}}{64}} \operatorname{atan}{\left(\frac{\sqrt{2} x}{\sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} - \frac{2 \sqrt{6} \sqrt{1 + \sqrt{3}}}{\sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} - \frac{\sqrt{2} \sqrt{1 + \sqrt{3}}}{\sqrt{\sqrt{3} + 2} \sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} + \frac{3 \sqrt{1 + \sqrt{3}}}{\sqrt{- 2 \sqrt{2} \sqrt{\sqrt{3} + 2} + 1 + 3 \sqrt{3}}} \right)}$$
Gráfica
Respuesta
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/ 4 2 / 3 2\\ / 4 2 / 3 2\\ - RootSum\3072*t + 320*t + 64*t + 3, t -> t*log\-10 - 960*t - 106*t + 96*t // + RootSum\3072*t + 320*t + 64*t + 3, t -> t*log\-9 - 960*t - 106*t + 96*t //
$$- \operatorname{RootSum} {\left(3072 t^{4} + 320 t^{2} + 64 t + 3, \left( t \mapsto t \log{\left(- 960 t^{3} + 96 t^{2} - 106 t - 10 \right)} \right)\right)} + \operatorname{RootSum} {\left(3072 t^{4} + 320 t^{2} + 64 t + 3, \left( t \mapsto t \log{\left(- 960 t^{3} + 96 t^{2} - 106 t - 9 \right)} \right)\right)}$$
=
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/ 4 2 / 3 2\\ / 4 2 / 3 2\\ - RootSum\3072*t + 320*t + 64*t + 3, t -> t*log\-10 - 960*t - 106*t + 96*t // + RootSum\3072*t + 320*t + 64*t + 3, t -> t*log\-9 - 960*t - 106*t + 96*t //
$$- \operatorname{RootSum} {\left(3072 t^{4} + 320 t^{2} + 64 t + 3, \left( t \mapsto t \log{\left(- 960 t^{3} + 96 t^{2} - 106 t - 10 \right)} \right)\right)} + \operatorname{RootSum} {\left(3072 t^{4} + 320 t^{2} + 64 t + 3, \left( t \mapsto t \log{\left(- 960 t^{3} + 96 t^{2} - 106 t - 9 \right)} \right)\right)}$$
-RootSum(3072*_t^4 + 320*_t^2 + 64*_t + 3, Lambda(_t, _t*log(-10 - 960*_t^3 - 106*_t + 96*_t^2))) + RootSum(3072*_t^4 + 320*_t^2 + 64*_t + 3, Lambda(_t, _t*log(-9 - 960*_t^3 - 106*_t + 96*_t^2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.