Integral de 1/((1+sinx-cosx)^2) dx
Solución
Respuesta (Indefinida)
[src]
/ /x\ 2/x\ / /x\\ / /x\\ /x\ 2/x\ / /x\\ / /x\\ /x\
| 3*tan|-| 2*tan |-|*log|tan|-|| 2*log|tan|-||*tan|-| 2*tan |-|*log|1 + tan|-|| 2*log|1 + tan|-||*tan|-|
| 1 1 \2/ \2/ \ \2// \ \2// \2/ \2/ \ \2// \ \2// \2/
| ---------------------- dx = C - -------------------- - -------------------- - --------------------- - -------------------- + ------------------------- + ------------------------
| 2 2/x\ /x\ 2/x\ /x\ 2/x\ /x\ 2/x\ /x\ 2/x\ /x\ 2/x\ /x\
| (1 + sin(x) - cos(x)) 2*tan |-| + 2*tan|-| 2*tan |-| + 2*tan|-| 2*tan |-| + 2*tan|-| 2*tan |-| + 2*tan|-| 2*tan |-| + 2*tan|-| 2*tan |-| + 2*tan|-|
| \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
/
$$\int \frac{1}{\left(\left(\sin{\left(x \right)} + 1\right) - \cos{\left(x \right)}\right)^{2}}\, dx = C + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)}} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} \right)} \tan{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)}} - \frac{3 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)}} - \frac{1}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)}}$$
$$\frac{2}{3} - \log{\left(\frac{3}{2} \right)}$$
=
$$\frac{2}{3} - \log{\left(\frac{3}{2} \right)}$$
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.