Sr Examen
Integral de cos²x/(1+cosx+sinx)² dx
Solución
Ha introducido
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2/3 / | | 2 | cos (x) | ---------------------- dx | 2 | (1 + cos(x) + sin(x)) | / 0
$$\int\limits_{0}^{\frac{2}{3}} \frac{\cos^{2}{\left(x \right)}}{\left(\left(\cos{\left(x \right)} + 1\right) + \sin{\left(x \right)}\right)^{2}}\, dx$$
Integral(cos(x)^2/(1 + cos(x) + sin(x))^2, (x, 0, 2/3))
Respuesta (Indefinida)
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/ | /x\ / 2/x\\ | 2 tan|-| log|1 + tan |-|| | cos (x) \2/ \ \2// | ---------------------- dx = C + ------ - ---------------- | 2 2 2 | (1 + cos(x) + sin(x)) | /
$$\int \frac{\cos^{2}{\left(x \right)}}{\left(\left(\cos{\left(x \right)} + 1\right) + \sin{\left(x \right)}\right)^{2}}\, dx = C - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{2} + \frac{\tan{\left(\frac{x}{2} \right)}}{2}$$
Gráfica
Respuesta
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/ 2 \ tan(1/3) log\1 + tan (1/3)/ -------- - ------------------ 2 2
$$- \frac{\log{\left(\tan^{2}{\left(\frac{1}{3} \right)} + 1 \right)}}{2} + \frac{\tan{\left(\frac{1}{3} \right)}}{2}$$
=
=
/ 2 \ tan(1/3) log\1 + tan (1/3)/ -------- - ------------------ 2 2
$$- \frac{\log{\left(\tan^{2}{\left(\frac{1}{3} \right)} + 1 \right)}}{2} + \frac{\tan{\left(\frac{1}{3} \right)}}{2}$$
tan(1/3)/2 - log(1 + tan(1/3)^2)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.