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