Sr Examen
Integral de (2-cos^3x)/(2*cos^2x-1)*sinx dx
Solución
Ha introducido
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1 / | | 3 | 2 - cos (x) | -------------*sin(x) dx | 2 | 2*cos (x) - 1 | / 0
$$\int\limits_{0}^{1} \frac{2 - \cos^{3}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} \sin{\left(x \right)}\, dx$$
Integral(((2 - cos(x)^3)/(2*cos(x)^2 - 1))*sin(x), (x, 0, 1))
Respuesta (Indefinida)
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/ // ___ / ___ \ \ | ||-\/ 2 *acoth\\/ 2 *cos(x)/ 2 | | 3 ||--------------------------- for cos (x) > 1/2| 2 / 2 \ | 2 - cos (x) || 2 | cos (x) log\-1 + 2*cos (x)/ | -------------*sin(x) dx = C - 2*|< | + ------- + ------------------- | 2 || ___ / ___ \ | 4 8 | 2*cos (x) - 1 ||-\/ 2 *atanh\\/ 2 *cos(x)/ 2 | | ||--------------------------- for cos (x) < 1/2| / \\ 2 /
$$\int \frac{2 - \cos^{3}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} \sin{\left(x \right)}\, dx = C - 2 \left(\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\sqrt{2} \cos{\left(x \right)} \right)}}{2} & \text{for}\: \cos^{2}{\left(x \right)} > \frac{1}{2} \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\sqrt{2} \cos{\left(x \right)} \right)}}{2} & \text{for}\: \cos^{2}{\left(x \right)} < \frac{1}{2} \end{cases}\right) + \frac{\log{\left(2 \cos^{2}{\left(x \right)} - 1 \right)}}{8} + \frac{\cos^{2}{\left(x \right)}}{4}$$
Gráfica
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.