log(u^2+1)/2+atan(u)=c-log(x) la ecuación
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Solución
Solución detallada
Tenemos la ecuación
$$\frac{\log{\left(u^{2} + 1 \right)}}{2} + \operatorname{atan}{\left(u \right)} = c - \log{\left(x \right)}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$\log{\left(x \right)} = c - \frac{\log{\left(u^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(u \right)}$$
Es la ecuación de la forma:
log(v)=p
Por definición log
v=e^p
entonces
$$x = e^{\frac{c - \frac{\log{\left(u^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(u \right)}}{1}}$$
simplificamos
$$x = \frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}$$
/ c - atan(u)\ / c - atan(u)\
|e | |e |
x1 = I*im|------------| + re|------------|
| ________ | | ________ |
| / 2 | | / 2 |
\\/ 1 + u / \\/ 1 + u /
$$x_{1} = \operatorname{re}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)} + i \operatorname{im}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)}$$
x1 = re(exp(c - atan(u))/sqrt(u^2 + 1)) + i*im(exp(c - atan(u))/sqrt(u^2 + 1))
Suma y producto de raíces
[src]
/ c - atan(u)\ / c - atan(u)\
|e | |e |
I*im|------------| + re|------------|
| ________ | | ________ |
| / 2 | | / 2 |
\\/ 1 + u / \\/ 1 + u /
$$\operatorname{re}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)} + i \operatorname{im}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)}$$
/ c - atan(u)\ / c - atan(u)\
|e | |e |
I*im|------------| + re|------------|
| ________ | | ________ |
| / 2 | | / 2 |
\\/ 1 + u / \\/ 1 + u /
$$\operatorname{re}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)} + i \operatorname{im}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)}$$
/ c - atan(u)\ / c - atan(u)\
|e | |e |
I*im|------------| + re|------------|
| ________ | | ________ |
| / 2 | | / 2 |
\\/ 1 + u / \\/ 1 + u /
$$\operatorname{re}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)} + i \operatorname{im}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)}$$
/ c - atan(u)\ / c - atan(u)\
|e | |e |
I*im|------------| + re|------------|
| ________ | | ________ |
| / 2 | | / 2 |
\\/ 1 + u / \\/ 1 + u /
$$\operatorname{re}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)} + i \operatorname{im}{\left(\frac{e^{c - \operatorname{atan}{\left(u \right)}}}{\sqrt{u^{2} + 1}}\right)}$$
i*im(exp(c - atan(u))/sqrt(1 + u^2)) + re(exp(c - atan(u))/sqrt(1 + u^2))