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