Sr Examen
Integral de sin^2x/(sinx+2cosx) dx
Solución
Ha introducido
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1 / | | 2 | sin (x) | ----------------- dx | sin(x) + 2*cos(x) | / 0
$$\int\limits_{0}^{1} \frac{\sin^{2}{\left(x \right)}}{\sin{\left(x \right)} + 2 \cos{\left(x \right)}}\, dx$$
Integral(sin(x)^2/(sin(x) + 2*cos(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ / ___ \ / ___ \ / ___ \ / ___ \ | /x\ ___ | 1 \/ 5 /x\| ___ | 1 \/ 5 /x\| ___ 2/x\ | 1 \/ 5 /x\| ___ 2/x\ | 1 \/ 5 /x\| | 2 20*tan|-| 4*\/ 5 *log|- - - ----- + tan|-|| 4*\/ 5 *log|- - + ----- + tan|-|| 4*\/ 5 *tan |-|*log|- - - ----- + tan|-|| 4*\/ 5 *tan |-|*log|- - + ----- + tan|-|| | sin (x) 10 \2/ \ 2 2 \2// \ 2 2 \2// \2/ \ 2 2 \2// \2/ \ 2 2 \2// | ----------------- dx = C - --------------- - --------------- - --------------------------------- + --------------------------------- - ----------------------------------------- + ----------------------------------------- | sin(x) + 2*cos(x) 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ | 25 + 25*tan |-| 25 + 25*tan |-| 25 + 25*tan |-| 25 + 25*tan |-| 25 + 25*tan |-| 25 + 25*tan |-| / \2/ \2/ \2/ \2/ \2/ \2/
$$\int \frac{\sin^{2}{\left(x \right)}}{\sin{\left(x \right)} + 2 \cos{\left(x \right)}}\, dx = C + \frac{4 \sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} - \frac{4 \sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{5}}{2} - \frac{1}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} - \frac{4 \sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{5}}{2} - \frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} - \frac{20 \tan{\left(\frac{x}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} - \frac{10}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25}$$
Gráfica
Respuesta
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/ ___\ / / ___\\ / / ___ \\ / ___ \ / / ___ \\ / ___ \
___ | 1 \/ 5 | ___ | |1 \/ 5 || ___ | |1 \/ 5 || ___ | 1 \/ 5 | ___ 2 | |1 \/ 5 || ___ 2 | 1 \/ 5 |
4*\/ 5 *log|- - + -----| 4*\/ 5 *|pi*I + log|- + -----|| 4*\/ 5 *|pi*I + log|- + ----- - tan(1/2)|| 4*\/ 5 *log|- - + ----- + tan(1/2)| 4*\/ 5 *tan (1/2)*|pi*I + log|- + ----- - tan(1/2)|| 4*\/ 5 *tan (1/2)*log|- - + ----- + tan(1/2)|
2 10 20*tan(1/2) \ 2 2 / \ \2 2 // \ \2 2 // \ 2 2 / \ \2 2 // \ 2 2 /
- - ----------------- - ----------------- - ------------------------ + ------------------------------- - ------------------------------------------ + ----------------------------------- - ---------------------------------------------------- + ---------------------------------------------
5 2 2 25 25 2 2 2 2
25 + 25*tan (1/2) 25 + 25*tan (1/2) 25 + 25*tan (1/2) 25 + 25*tan (1/2) 25 + 25*tan (1/2) 25 + 25*tan (1/2)
$$- \frac{20 \tan{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{10}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{5}}{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{5}}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{25} + \frac{2}{5} - \frac{4 \sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{4 \sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{25}$$
=
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/ ___\ / / ___\\ / / ___ \\ / ___ \ / / ___ \\ / ___ \
___ | 1 \/ 5 | ___ | |1 \/ 5 || ___ | |1 \/ 5 || ___ | 1 \/ 5 | ___ 2 | |1 \/ 5 || ___ 2 | 1 \/ 5 |
4*\/ 5 *log|- - + -----| 4*\/ 5 *|pi*I + log|- + -----|| 4*\/ 5 *|pi*I + log|- + ----- - tan(1/2)|| 4*\/ 5 *log|- - + ----- + tan(1/2)| 4*\/ 5 *tan (1/2)*|pi*I + log|- + ----- - tan(1/2)|| 4*\/ 5 *tan (1/2)*log|- - + ----- + tan(1/2)|
2 10 20*tan(1/2) \ 2 2 / \ \2 2 // \ \2 2 // \ 2 2 / \ \2 2 // \ 2 2 /
- - ----------------- - ----------------- - ------------------------ + ------------------------------- - ------------------------------------------ + ----------------------------------- - ---------------------------------------------------- + ---------------------------------------------
5 2 2 25 25 2 2 2 2
25 + 25*tan (1/2) 25 + 25*tan (1/2) 25 + 25*tan (1/2) 25 + 25*tan (1/2) 25 + 25*tan (1/2) 25 + 25*tan (1/2)
$$- \frac{20 \tan{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{10}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{5}}{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{5}}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{25} + \frac{2}{5} - \frac{4 \sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{4 \sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{25}$$
2/5 - 10/(25 + 25*tan(1/2)^2) - 20*tan(1/2)/(25 + 25*tan(1/2)^2) - 4*sqrt(5)*log(-1/2 + sqrt(5)/2)/25 + 4*sqrt(5)*(pi*i + log(1/2 + sqrt(5)/2))/25 - 4*sqrt(5)*(pi*i + log(1/2 + sqrt(5)/2 - tan(1/2)))/(25 + 25*tan(1/2)^2) + 4*sqrt(5)*log(-1/2 + sqrt(5)/2 + tan(1/2))/(25 + 25*tan(1/2)^2) - 4*sqrt(5)*tan(1/2)^2*(pi*i + log(1/2 + sqrt(5)/2 - tan(1/2)))/(25 + 25*tan(1/2)^2) + 4*sqrt(5)*tan(1/2)^2*log(-1/2 + sqrt(5)/2 + tan(1/2))/(25 + 25*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.