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