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