Sr Examen
Integral de cos(x)*ln(cos(x)) dx
Solución
Ha introducido
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1 / | | cos(x)*log(cos(x)) dx | / 0
$$\int\limits_{0}^{1} \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\, dx$$
Integral(cos(x)*log(cos(x)), (x, 0, 1))
Solución detallada
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Usamos la integración por partes:
que y que .
Entonces .
Para buscar :
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La integral del coseno es seno:
Ahora resolvemos podintegral.
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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Añadimos la constante de integración:
Respuesta:
Respuesta (Indefinida)
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/ | log(1 + sin(x)) log(-1 + sin(x)) | cos(x)*log(cos(x)) dx = C + --------------- - sin(x) - ---------------- + log(cos(x))*sin(x) | 2 2 /
$$\int \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\, dx = C - \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} + \log{\left(\cos{\left(x \right)} \right)} \sin{\left(x \right)} - \sin{\left(x \right)}$$
Gráfica
Respuesta
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/ 2 \ / 2 \ / 2 \
| 1 tan (1/2) | 2 | 1 tan (1/2) | | 1 tan (1/2) |
log|------------- - -------------| tan (1/2)*log|------------- - -------------| 2*log|------------- - -------------|*tan(1/2)
/ 2 \ | 2 2 | 2 / 2 \ | 2 2 | 2 | 2 2 |
log\1 + tan (1/2)/ \1 + tan (1/2) 1 + tan (1/2)/ 2*tan(1/2) 2*log(1 + tan(1/2)) tan (1/2)*log\1 + tan (1/2)/ \1 + tan (1/2) 1 + tan (1/2)/ 2*tan (1/2)*log(1 + tan(1/2)) \1 + tan (1/2) 1 + tan (1/2)/
- ------------------ - ---------------------------------- - ------------- + ------------------- - ---------------------------- - -------------------------------------------- + ----------------------------- + ---------------------------------------------
2 2 2 2 2 2 2 2
1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2)
$$- \frac{2 \tan{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1}$$
=
=
/ 2 \ / 2 \ / 2 \
| 1 tan (1/2) | 2 | 1 tan (1/2) | | 1 tan (1/2) |
log|------------- - -------------| tan (1/2)*log|------------- - -------------| 2*log|------------- - -------------|*tan(1/2)
/ 2 \ | 2 2 | 2 / 2 \ | 2 2 | 2 | 2 2 |
log\1 + tan (1/2)/ \1 + tan (1/2) 1 + tan (1/2)/ 2*tan(1/2) 2*log(1 + tan(1/2)) tan (1/2)*log\1 + tan (1/2)/ \1 + tan (1/2) 1 + tan (1/2)/ 2*tan (1/2)*log(1 + tan(1/2)) \1 + tan (1/2) 1 + tan (1/2)/
- ------------------ - ---------------------------------- - ------------- + ------------------- - ---------------------------- - -------------------------------------------- + ----------------------------- + ---------------------------------------------
2 2 2 2 2 2 2 2
1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2)
$$- \frac{2 \tan{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(- \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{1}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1}$$
-log(1 + tan(1/2)^2)/(1 + tan(1/2)^2) - log(1/(1 + tan(1/2)^2) - tan(1/2)^2/(1 + tan(1/2)^2))/(1 + tan(1/2)^2) - 2*tan(1/2)/(1 + tan(1/2)^2) + 2*log(1 + tan(1/2))/(1 + tan(1/2)^2) - tan(1/2)^2*log(1 + tan(1/2)^2)/(1 + tan(1/2)^2) - tan(1/2)^2*log(1/(1 + tan(1/2)^2) - tan(1/2)^2/(1 + tan(1/2)^2))/(1 + tan(1/2)^2) + 2*tan(1/2)^2*log(1 + tan(1/2))/(1 + tan(1/2)^2) + 2*log(1/(1 + tan(1/2)^2) - tan(1/2)^2/(1 + tan(1/2)^2))*tan(1/2)/(1 + tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.