Sr Examen
Resolución de integrales/
(2sin(x)^3)/((cos(x)+1)^3)-(sin(x))/(cos(x)+1)+(3cos(x)*sin(x))/((cos(x)+1)^2)
Integral de (2sin(x)^3)/((cos(x)+1)^3)-(sin(x))/(cos(x)+1)+(3cos(x)*sin(x))/((cos(x)+1)^2) dx
Solución
Ha introducido
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1 / | | / 3 \ | | 2*sin (x) sin(x) 3*cos(x)*sin(x)| | |------------- - ---------- + ---------------| dx | | 3 cos(x) + 1 2 | | \(cos(x) + 1) (cos(x) + 1) / | / 0
$$\int\limits_{0}^{1} \left(\frac{\sin{\left(x \right)} 3 \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \left(\frac{2 \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)\right)\, dx$$
Integral((2*sin(x)^3)/(cos(x) + 1)^3 - sin(x)/(cos(x) + 1) + ((3*cos(x))*sin(x))/(cos(x) + 1)^2, (x, 0, 1))
Respuesta (Indefinida)
[src]
/ | | / 3 \ / 2 \ 2 2 | | 2*sin (x) sin(x) 3*cos(x)*sin(x)| 3 4 3*log\1 + cos (x) + 2*cos(x)/ 2*sin (x) 4*cos(x) 4*log(1 + cos(x)) 4*cos (x)*log(1 + cos(x)) 8*cos(x)*log(1 + cos(x)) | |------------- - ---------- + ---------------| dx = C - ---------- + ------------------------ - ----------------------------- + ------------------------ + ------------------------ + ------------------------ + ------------------------- + ------------------------ + log(cos(x) + 1) | | 3 cos(x) + 1 2 | 1 + cos(x) 2 2 2 2 2 2 2 | \(cos(x) + 1) (cos(x) + 1) / 2 + 2*cos (x) + 4*cos(x) 2 + 2*cos (x) + 4*cos(x) 2 + 2*cos (x) + 4*cos(x) 2 + 2*cos (x) + 4*cos(x) 2 + 2*cos (x) + 4*cos(x) 2 + 2*cos (x) + 4*cos(x) | /
$$\int \left(\frac{\sin{\left(x \right)} 3 \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \left(\frac{2 \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)\right)\, dx = C + \log{\left(\cos{\left(x \right)} + 1 \right)} - \frac{3 \log{\left(\cos^{2}{\left(x \right)} + 2 \cos{\left(x \right)} + 1 \right)}}{2} + \frac{4 \log{\left(\cos{\left(x \right)} + 1 \right)} \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{8 \log{\left(\cos{\left(x \right)} + 1 \right)} \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{4 \log{\left(\cos{\left(x \right)} + 1 \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{2 \sin^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{4 \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} + \frac{4}{2 \cos^{2}{\left(x \right)} + 4 \cos{\left(x \right)} + 2} - \frac{3}{\cos{\left(x \right)} + 1}$$
Gráfica
Respuesta
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2 2
1 3 4 3*log(1 + cos(1)) 2*sin (1) 4*cos(1) 4*log(1 + cos(1)) 3*cos(1)*log(1 + cos(1)) 4*cos (1)*log(1 + cos(1)) 8*cos(1)*log(1 + cos(1))
- - ---------- + ------------------------ - ----------------- + ------------------------ + ------------------------ + ------------------------ - ------------------------ + ------------------------- + ------------------------ + log(1 + cos(1))
2 1 + cos(1) 2 1 + cos(1) 2 2 2 1 + cos(1) 2 2
2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1)
$$- \frac{3}{\cos{\left(1 \right)} + 1} - \frac{3 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{\cos{\left(1 \right)} + 1} - \frac{3 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{\cos{\left(1 \right)} + 1} + \frac{4 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{2 \sin^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{4 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{8 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \log{\left(\cos{\left(1 \right)} + 1 \right)} + \frac{4 \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{1}{2} + \frac{4}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}}$$
=
=
2 2
1 3 4 3*log(1 + cos(1)) 2*sin (1) 4*cos(1) 4*log(1 + cos(1)) 3*cos(1)*log(1 + cos(1)) 4*cos (1)*log(1 + cos(1)) 8*cos(1)*log(1 + cos(1))
- - ---------- + ------------------------ - ----------------- + ------------------------ + ------------------------ + ------------------------ - ------------------------ + ------------------------- + ------------------------ + log(1 + cos(1))
2 1 + cos(1) 2 1 + cos(1) 2 2 2 1 + cos(1) 2 2
2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1)
$$- \frac{3}{\cos{\left(1 \right)} + 1} - \frac{3 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{\cos{\left(1 \right)} + 1} - \frac{3 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{\cos{\left(1 \right)} + 1} + \frac{4 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{2 \sin^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{4 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{8 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \log{\left(\cos{\left(1 \right)} + 1 \right)} + \frac{4 \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} + \frac{1}{2} + \frac{4}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}}$$
1/2 - 3/(1 + cos(1)) + 4/(2 + 2*cos(1)^2 + 4*cos(1)) - 3*log(1 + cos(1))/(1 + cos(1)) + 2*sin(1)^2/(2 + 2*cos(1)^2 + 4*cos(1)) + 4*cos(1)/(2 + 2*cos(1)^2 + 4*cos(1)) + 4*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) - 3*cos(1)*log(1 + cos(1))/(1 + cos(1)) + 4*cos(1)^2*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) + 8*cos(1)*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) + log(1 + cos(1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.