Sr Examen
Integral de tan(x/2)/(1-tan(x/2)^2) dx
Solución
Ha introducido
[src]
1 / | | /x\ | tan|-| | \2/ | ----------- dx | 2/x\ | 1 - tan |-| | \2/ | / 0
$$\int\limits_{0}^{1} \frac{\tan{\left(\frac{x}{2} \right)}}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}\, dx$$
Integral(tan(x/2)/(1 - tan(x/2)^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ // / 2/x\\ \ | ||-acoth|tan |-|| | | /x\ || \ \2// 4/x\ | | tan|-| ||---------------- for tan |-| > 1| | \2/ || 2 \2/ | | ----------- dx = C - 2*|< | | 2/x\ || / 2/x\\ | | 1 - tan |-| ||-atanh|tan |-|| | | \2/ || \ \2// 4/x\ | | ||---------------- for tan |-| < 1| / \\ 2 \2/ /
$$\int \frac{\tan{\left(\frac{x}{2} \right)}}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}\, dx = C - 2 \left(\begin{cases} - \frac{\operatorname{acoth}{\left(\tan^{2}{\left(\frac{x}{2} \right)} \right)}}{2} & \text{for}\: \tan^{4}{\left(\frac{x}{2} \right)} > 1 \\- \frac{\operatorname{atanh}{\left(\tan^{2}{\left(\frac{x}{2} \right)} \right)}}{2} & \text{for}\: \tan^{4}{\left(\frac{x}{2} \right)} < 1 \end{cases}\right)$$
Gráfica
Respuesta
[src]
/ 2 \
log\1 + tan (1/2)/ log(1 - tan(1/2)) log(1 + tan(1/2))
------------------ - ----------------- - -----------------
2 2 2
$$- \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)}}{2}$$
=
=
/ 2 \
log\1 + tan (1/2)/ log(1 - tan(1/2)) log(1 + tan(1/2))
------------------ - ----------------- - -----------------
2 2 2
$$- \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)}}{2}$$
log(1 + tan(1/2)^2)/2 - log(1 - tan(1/2))/2 - log(1 + tan(1/2))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.