Sr Examen
Integral de 1/(cos(x)*(1+cos(x))) dx
Solución
Ha introducido
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1 / | | 1 | ------------------- dx | cos(x)*(1 + cos(x)) | / pi -- 4
$$\int\limits_{\frac{\pi}{4}}^{1} \frac{1}{\left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)}}\, dx$$
Integral(1/(cos(x)*(1 + cos(x))), (x, pi/4, 1))
Respuesta (Indefinida)
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/ | | 1 / /x\\ /x\ / /x\\ | ------------------- dx = C - log|-1 + tan|-|| - tan|-| + log|1 + tan|-|| | cos(x)*(1 + cos(x)) \ \2// \2/ \ \2// | /
$$\int \frac{1}{\left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)}}\, dx = C - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \tan{\left(\frac{x}{2} \right)}$$
Gráfica
Respuesta
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___ / ___\ / ___\ -1 + \/ 2 - log\\/ 2 / - log(1 - tan(1/2)) - tan(1/2) + log(1 + tan(1/2)) + log\2 - \/ 2 /
$$-1 - \tan{\left(\frac{1}{2} \right)} + \log{\left(2 - \sqrt{2} \right)} - \log{\left(\sqrt{2} \right)} + \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} - \log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + \sqrt{2}$$
=
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___ / ___\ / ___\ -1 + \/ 2 - log\\/ 2 / - log(1 - tan(1/2)) - tan(1/2) + log(1 + tan(1/2)) + log\2 - \/ 2 /
$$-1 - \tan{\left(\frac{1}{2} \right)} + \log{\left(2 - \sqrt{2} \right)} - \log{\left(\sqrt{2} \right)} + \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} - \log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + \sqrt{2}$$
-1 + sqrt(2) - log(sqrt(2)) - log(1 - tan(1/2)) - tan(1/2) + log(1 + tan(1/2)) + log(2 - sqrt(2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.