Sr Examen
Resolución de integrales/
(24*sin(x)^5)/((cos(x)+1)^5)-((20*sin(x)^3)/((cos(x)+1)^3))+(sin(x))/(cos(x)+1)-(15*cos(x)*sin(x))/((cos(x)+1)^2)
Integral de (24*sin(x)^5)/((cos(x)+1)^5)-((20*sin(x)^3)/((cos(x)+1)^3))+(sin(x))/(cos(x)+1)-(15*cos(x)*sin(x))/((cos(x)+1)^2) dx
Solución
Ha introducido
[src]
1 / | | / 5 3 \ | | 24*sin (x) 20*sin (x) sin(x) 15*cos(x)*sin(x)| | |------------- - ------------- + ---------- - ----------------| dx | | 5 3 cos(x) + 1 2 | | \(cos(x) + 1) (cos(x) + 1) (cos(x) + 1) / | / 0
$$\int\limits_{0}^{1} \left(- \frac{\sin{\left(x \right)} 15 \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \left(\left(\frac{24 \sin^{5}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{5}} - \frac{20 \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}}\right) + \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)\right)\, dx$$
Integral((24*sin(x)^5)/(cos(x) + 1)^5 - 20*sin(x)^3/(cos(x) + 1)^3 + sin(x)/(cos(x) + 1) - (15*cos(x))*sin(x)/(cos(x) + 1)^2, (x, 0, 1))
Gráfica
Respuesta
[src]
2 3 2 2 4 2 3 4 2 2 2 2
41 384 40 15 1344*cos (1) 1248*cos(1) 480*cos (1) 288*log(1 + cos(1)) 48*sin (1) 40*cos(1) 40*log(1 + cos(1)) 20*sin (1) 15*log(1 + cos(1)) 72*sin (1) 1728*cos (1)*log(1 + cos(1)) 1152*cos (1)*log(1 + cos(1)) 1152*cos(1)*log(1 + cos(1)) 288*cos (1)*log(1 + cos(1)) 192*sin (1)*cos(1) 144*cos (1)*sin (1) 80*cos(1)*log(1 + cos(1)) 40*cos (1)*log(1 + cos(1)) 15*cos(1)*log(1 + cos(1))
-- - log(1 + cos(1)) - ----------------------------------------------------- - ------------------------ + ---------- + 30*log(2) - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ------------------------ - ------------------------ - ------------------------ + ------------------ + ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ------------------------- - -------------------------- + -------------------------
2 4 3 2 2 1 + cos(1) 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 2 2 2 1 + cos(1) 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 2 2 1 + cos(1)
12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 2 + 2*cos (1) + 4*cos(1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1)
$$- \frac{1248 \cos{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{40}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{1344 \cos^{2}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{384}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{40 \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{1152 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{80 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{40 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{1728 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{2}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{20 \sin^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{288 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{1152 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{3}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{480 \cos^{3}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{192 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{40 \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{48 \sin^{2}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{144 \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \log{\left(\cos{\left(1 \right)} + 1 \right)} - \frac{288 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{4}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} + \frac{72 \sin^{4}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} + \frac{15 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{\cos{\left(1 \right)} + 1} + \frac{15 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{\cos{\left(1 \right)} + 1} + \frac{15}{\cos{\left(1 \right)} + 1} + \frac{41}{2} + 30 \log{\left(2 \right)}$$
=
=
2 3 2 2 4 2 3 4 2 2 2 2
41 384 40 15 1344*cos (1) 1248*cos(1) 480*cos (1) 288*log(1 + cos(1)) 48*sin (1) 40*cos(1) 40*log(1 + cos(1)) 20*sin (1) 15*log(1 + cos(1)) 72*sin (1) 1728*cos (1)*log(1 + cos(1)) 1152*cos (1)*log(1 + cos(1)) 1152*cos(1)*log(1 + cos(1)) 288*cos (1)*log(1 + cos(1)) 192*sin (1)*cos(1) 144*cos (1)*sin (1) 80*cos(1)*log(1 + cos(1)) 40*cos (1)*log(1 + cos(1)) 15*cos(1)*log(1 + cos(1))
-- - log(1 + cos(1)) - ----------------------------------------------------- - ------------------------ + ---------- + 30*log(2) - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ------------------------ - ------------------------ - ------------------------ + ------------------ + ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ----------------------------------------------------- - ------------------------- - -------------------------- + -------------------------
2 4 3 2 2 1 + cos(1) 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 2 2 2 1 + cos(1) 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 2 2 1 + cos(1)
12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 2 + 2*cos (1) + 4*cos(1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 12 + 12*cos (1) + 48*cos (1) + 48*cos(1) + 72*cos (1) 2 + 2*cos (1) + 4*cos(1) 2 + 2*cos (1) + 4*cos(1)
$$- \frac{1248 \cos{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{40}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{1344 \cos^{2}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{384}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{40 \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{1152 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{80 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{40 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{1728 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{2}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{20 \sin^{2}{\left(1 \right)}}{2 \cos^{2}{\left(1 \right)} + 2 + 4 \cos{\left(1 \right)}} - \frac{288 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{1152 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{3}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{480 \cos^{3}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{192 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{40 \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{48 \sin^{2}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \frac{144 \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} - \log{\left(\cos{\left(1 \right)} + 1 \right)} - \frac{288 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos^{4}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} + \frac{72 \sin^{4}{\left(1 \right)}}{12 \cos^{4}{\left(1 \right)} + 48 \cos^{3}{\left(1 \right)} + 12 + 72 \cos^{2}{\left(1 \right)} + 48 \cos{\left(1 \right)}} + \frac{15 \log{\left(\cos{\left(1 \right)} + 1 \right)} \cos{\left(1 \right)}}{\cos{\left(1 \right)} + 1} + \frac{15 \log{\left(\cos{\left(1 \right)} + 1 \right)}}{\cos{\left(1 \right)} + 1} + \frac{15}{\cos{\left(1 \right)} + 1} + \frac{41}{2} + 30 \log{\left(2 \right)}$$
41/2 - log(1 + cos(1)) - 384/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 40/(2 + 2*cos(1)^2 + 4*cos(1)) + 15/(1 + cos(1)) + 30*log(2) - 1344*cos(1)^2/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 1248*cos(1)/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 480*cos(1)^3/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 288*log(1 + cos(1))/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 48*sin(1)^2/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 40*cos(1)/(2 + 2*cos(1)^2 + 4*cos(1)) - 40*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) - 20*sin(1)^2/(2 + 2*cos(1)^2 + 4*cos(1)) + 15*log(1 + cos(1))/(1 + cos(1)) + 72*sin(1)^4/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 1728*cos(1)^2*log(1 + cos(1))/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 1152*cos(1)^3*log(1 + cos(1))/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 1152*cos(1)*log(1 + cos(1))/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 288*cos(1)^4*log(1 + cos(1))/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 192*sin(1)^2*cos(1)/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 144*cos(1)^2*sin(1)^2/(12 + 12*cos(1)^4 + 48*cos(1)^3 + 48*cos(1) + 72*cos(1)^2) - 80*cos(1)*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) - 40*cos(1)^2*log(1 + cos(1))/(2 + 2*cos(1)^2 + 4*cos(1)) + 15*cos(1)*log(1 + cos(1))/(1 + cos(1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.