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