Sr Examen
Integral de (cosx)^2/(2*sinx+3*cosx) dx
Solución
Ha introducido
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1 / | | 2 | cos (x) | ------------------- dx | 2*sin(x) + 3*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{\cos^{2}{\left(x \right)}}{2 \sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx$$
Integral(cos(x)^2/(2*sin(x) + 3*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ / ____ \ / ____ \ / ____ \ / ____ \ | /x\ ____ | 2 \/ 13 /x\| ____ | 2 \/ 13 /x\| ____ 2/x\ | 2 \/ 13 /x\| ____ 2/x\ | 2 \/ 13 /x\| | 2 78*tan|-| 4*\/ 13 *log|- - - ------ + tan|-|| 4*\/ 13 *log|- - + ------ + tan|-|| 4*\/ 13 *tan |-|*log|- - - ------ + tan|-|| 4*\/ 13 *tan |-|*log|- - + ------ + tan|-|| | cos (x) 52 \2/ \ 3 3 \2// \ 3 3 \2// \2/ \ 3 3 \2// \2/ \ 3 3 \2// | ------------------- dx = C + ----------------- + ----------------- - ----------------------------------- + ----------------------------------- - ------------------------------------------- + ------------------------------------------- | 2*sin(x) + 3*cos(x) 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ | 169 + 169*tan |-| 169 + 169*tan |-| 169 + 169*tan |-| 169 + 169*tan |-| 169 + 169*tan |-| 169 + 169*tan |-| / \2/ \2/ \2/ \2/ \2/ \2/
$$\int \frac{\cos^{2}{\left(x \right)}}{2 \sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx = C + \frac{4 \sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} + \frac{4 \sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} - \frac{4 \sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{13}}{3} - \frac{2}{3} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} - \frac{4 \sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{13}}{3} - \frac{2}{3} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} + \frac{78 \tan{\left(\frac{x}{2} \right)}}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169} + \frac{52}{169 \tan^{2}{\left(\frac{x}{2} \right)} + 169}$$
Gráfica
Respuesta
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/ ____\ / / ____\\ / / ____\\ / ____ \ / / ____\\ / ____ \
____ | 2 \/ 13 | ____ | |2 \/ 13 || ____ | |2 \/ 13 || ____ | 2 \/ 13 | ____ 2 | |2 \/ 13 || ____ 2 | 2 \/ 13 |
4*\/ 13 *log|- - + ------| 4*\/ 13 *|pi*I + log|- + ------|| 4*\/ 13 *|pi*I + log|- - tan(1/2) + ------|| 4*\/ 13 *log|- - + ------ + tan(1/2)| 4*\/ 13 *tan (1/2)*|pi*I + log|- - tan(1/2) + ------|| 4*\/ 13 *tan (1/2)*log|- - + ------ + tan(1/2)|
4 52 78*tan(1/2) \ 3 3 / \ \3 3 // \ \3 3 // \ 3 3 / \ \3 3 // \ 3 3 /
- -- + ------------------- + ------------------- - -------------------------- + --------------------------------- - -------------------------------------------- + ------------------------------------- - ------------------------------------------------------ + -----------------------------------------------
13 2 2 169 169 2 2 2 2
169 + 169*tan (1/2) 169 + 169*tan (1/2) 169 + 169*tan (1/2) 169 + 169*tan (1/2) 169 + 169*tan (1/2) 169 + 169*tan (1/2)
$$- \frac{4}{13} + \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{169} + \frac{78 \tan{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{52}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{4 \sqrt{13} \left(\log{\left(\frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{169}$$
=
=
/ ____\ / / ____\\ / / ____\\ / ____ \ / / ____\\ / ____ \
____ | 2 \/ 13 | ____ | |2 \/ 13 || ____ | |2 \/ 13 || ____ | 2 \/ 13 | ____ 2 | |2 \/ 13 || ____ 2 | 2 \/ 13 |
4*\/ 13 *log|- - + ------| 4*\/ 13 *|pi*I + log|- + ------|| 4*\/ 13 *|pi*I + log|- - tan(1/2) + ------|| 4*\/ 13 *log|- - + ------ + tan(1/2)| 4*\/ 13 *tan (1/2)*|pi*I + log|- - tan(1/2) + ------|| 4*\/ 13 *tan (1/2)*log|- - + ------ + tan(1/2)|
4 52 78*tan(1/2) \ 3 3 / \ \3 3 // \ \3 3 // \ 3 3 / \ \3 3 // \ 3 3 /
- -- + ------------------- + ------------------- - -------------------------- + --------------------------------- - -------------------------------------------- + ------------------------------------- - ------------------------------------------------------ + -----------------------------------------------
13 2 2 169 169 2 2 2 2
169 + 169*tan (1/2) 169 + 169*tan (1/2) 169 + 169*tan (1/2) 169 + 169*tan (1/2) 169 + 169*tan (1/2) 169 + 169*tan (1/2)
$$- \frac{4}{13} + \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{3} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \log{\left(- \frac{2}{3} + \frac{\sqrt{13}}{3} \right)}}{169} + \frac{78 \tan{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{52}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} - \frac{4 \sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{169 \tan^{2}{\left(\frac{1}{2} \right)} + 169} + \frac{4 \sqrt{13} \left(\log{\left(\frac{2}{3} + \frac{\sqrt{13}}{3} \right)} + i \pi\right)}{169}$$
-4/13 + 52/(169 + 169*tan(1/2)^2) + 78*tan(1/2)/(169 + 169*tan(1/2)^2) - 4*sqrt(13)*log(-2/3 + sqrt(13)/3)/169 + 4*sqrt(13)*(pi*i + log(2/3 + sqrt(13)/3))/169 - 4*sqrt(13)*(pi*i + log(2/3 - tan(1/2) + sqrt(13)/3))/(169 + 169*tan(1/2)^2) + 4*sqrt(13)*log(-2/3 + sqrt(13)/3 + tan(1/2))/(169 + 169*tan(1/2)^2) - 4*sqrt(13)*tan(1/2)^2*(pi*i + log(2/3 - tan(1/2) + sqrt(13)/3))/(169 + 169*tan(1/2)^2) + 4*sqrt(13)*tan(1/2)^2*log(-2/3 + sqrt(13)/3 + tan(1/2))/(169 + 169*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.