Sr Examen
Integral de cosx^2/(sinx+3cosx) dx
Solución
Ha introducido
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1 / | | 2 | cos (x) | ----------------- dx | sin(x) + 3*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx$$
Integral(cos(x)^2/(sin(x) + 3*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ / ____ \ / ____ \ / ____ \ / ____ \ | /x\ ____ | 1 \/ 10 /x\| ____ | 1 \/ 10 /x\| ____ 2/x\ | 1 \/ 10 /x\| ____ 2/x\ | 1 \/ 10 /x\| | 2 60*tan|-| \/ 10 *log|- - + ------ + tan|-|| \/ 10 *log|- - - ------ + tan|-|| \/ 10 *tan |-|*log|- - + ------ + tan|-|| \/ 10 *tan |-|*log|- - - ------ + tan|-|| | cos (x) 20 \2/ \ 3 3 \2// \ 3 3 \2// \2/ \ 3 3 \2// \2/ \ 3 3 \2// | ----------------- dx = C + ----------------- + ----------------- + --------------------------------- - --------------------------------- + ----------------------------------------- - ----------------------------------------- | sin(x) + 3*cos(x) 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ | 100 + 100*tan |-| 100 + 100*tan |-| 100 + 100*tan |-| 100 + 100*tan |-| 100 + 100*tan |-| 100 + 100*tan |-| / \2/ \2/ \2/ \2/ \2/ \2/
$$\int \frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 3 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} + \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{3} + \frac{\sqrt{10}}{3} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} - \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{10}}{3} - \frac{1}{3} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} - \frac{\sqrt{10} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{10}}{3} - \frac{1}{3} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} + \frac{60 \tan{\left(\frac{x}{2} \right)}}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100} + \frac{20}{100 \tan^{2}{\left(\frac{x}{2} \right)} + 100}$$
Gráfica
Respuesta
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/ ____\ / / ____\\ / ____ \ / / ____\\ / ____ \ / / ____\\
____ | 1 \/ 10 | ____ | |1 \/ 10 || ____ | 1 \/ 10 | ____ | |1 \/ 10 || ____ 2 | 1 \/ 10 | ____ 2 | |1 \/ 10 ||
\/ 10 *log|- - + ------| \/ 10 *|pi*I + log|- + ------|| \/ 10 *log|- - + ------ + tan(1/2)| \/ 10 *|pi*I + log|- - tan(1/2) + ------|| \/ 10 *tan (1/2)*log|- - + ------ + tan(1/2)| \/ 10 *tan (1/2)*|pi*I + log|- - tan(1/2) + ------||
1 20 60*tan(1/2) \ 3 3 / \ \3 3 // \ 3 3 / \ \3 3 // \ 3 3 / \ \3 3 //
- - + ------------------- + ------------------- - ------------------------ + ------------------------------- + ----------------------------------- - ------------------------------------------ + --------------------------------------------- - ----------------------------------------------------
5 2 2 100 100 2 2 2 2
100 + 100*tan (1/2) 100 + 100*tan (1/2) 100 + 100*tan (1/2) 100 + 100*tan (1/2) 100 + 100*tan (1/2) 100 + 100*tan (1/2)
$$- \frac{1}{5} + \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{10}}{3} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{10}}{3} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \frac{\sqrt{10}}{3} \right)}}{100} + \frac{20}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{60 \tan{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right)}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{\sqrt{10} \left(\log{\left(\frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right)}{100}$$
=
=
/ ____\ / / ____\\ / ____ \ / / ____\\ / ____ \ / / ____\\
____ | 1 \/ 10 | ____ | |1 \/ 10 || ____ | 1 \/ 10 | ____ | |1 \/ 10 || ____ 2 | 1 \/ 10 | ____ 2 | |1 \/ 10 ||
\/ 10 *log|- - + ------| \/ 10 *|pi*I + log|- + ------|| \/ 10 *log|- - + ------ + tan(1/2)| \/ 10 *|pi*I + log|- - tan(1/2) + ------|| \/ 10 *tan (1/2)*log|- - + ------ + tan(1/2)| \/ 10 *tan (1/2)*|pi*I + log|- - tan(1/2) + ------||
1 20 60*tan(1/2) \ 3 3 / \ \3 3 // \ 3 3 / \ \3 3 // \ 3 3 / \ \3 3 //
- - + ------------------- + ------------------- - ------------------------ + ------------------------------- + ----------------------------------- - ------------------------------------------ + --------------------------------------------- - ----------------------------------------------------
5 2 2 100 100 2 2 2 2
100 + 100*tan (1/2) 100 + 100*tan (1/2) 100 + 100*tan (1/2) 100 + 100*tan (1/2) 100 + 100*tan (1/2) 100 + 100*tan (1/2)
$$- \frac{1}{5} + \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{10}}{3} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{10}}{3} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \log{\left(- \frac{1}{3} + \frac{\sqrt{10}}{3} \right)}}{100} + \frac{20}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{60 \tan{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right)}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} - \frac{\sqrt{10} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{100 \tan^{2}{\left(\frac{1}{2} \right)} + 100} + \frac{\sqrt{10} \left(\log{\left(\frac{1}{3} + \frac{\sqrt{10}}{3} \right)} + i \pi\right)}{100}$$
-1/5 + 20/(100 + 100*tan(1/2)^2) + 60*tan(1/2)/(100 + 100*tan(1/2)^2) - sqrt(10)*log(-1/3 + sqrt(10)/3)/100 + sqrt(10)*(pi*i + log(1/3 + sqrt(10)/3))/100 + sqrt(10)*log(-1/3 + sqrt(10)/3 + tan(1/2))/(100 + 100*tan(1/2)^2) - sqrt(10)*(pi*i + log(1/3 - tan(1/2) + sqrt(10)/3))/(100 + 100*tan(1/2)^2) + sqrt(10)*tan(1/2)^2*log(-1/3 + sqrt(10)/3 + tan(1/2))/(100 + 100*tan(1/2)^2) - sqrt(10)*tan(1/2)^2*(pi*i + log(1/3 - tan(1/2) + sqrt(10)/3))/(100 + 100*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.