Sr Examen
Integral de cos^4(x)sin^2(x)x dx
Solución
Ha introducido
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1 / | | 4 2 | cos (x)*sin (x)*x dx | / 0
$$\int\limits_{0}^{1} x \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$
Integral((cos(x)^4*sin(x)^2)*x, (x, 0, 1))
Respuesta (Indefinida)
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/ | 6 6 4 2 2 6 2 6 5 3 3 5 2 2 4 2 4 2 | 4 2 sin (x) cos (x) cos (x)*sin (x) x *cos (x) x *sin (x) x*cos (x)*sin(x) x*cos (x)*sin (x) x*sin (x)*cos(x) 3*x *cos (x)*sin (x) 3*x *cos (x)*sin (x) | cos (x)*sin (x)*x dx = C - ------- + ------- + --------------- + ---------- + ---------- - ---------------- + ----------------- + ---------------- + -------------------- + -------------------- | 96 288 24 32 32 16 6 16 32 32 /
$$\int x \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = C + \frac{x^{2} \sin^{6}{\left(x \right)}}{32} + \frac{3 x^{2} \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{32} + \frac{3 x^{2} \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{32} + \frac{x^{2} \cos^{6}{\left(x \right)}}{32} + \frac{x \sin^{5}{\left(x \right)} \cos{\left(x \right)}}{16} + \frac{x \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)}}{6} - \frac{x \sin{\left(x \right)} \cos^{5}{\left(x \right)}}{16} - \frac{\sin^{6}{\left(x \right)}}{96} + \frac{\sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{24} + \frac{\cos^{6}{\left(x \right)}}{288}$$
Gráfica
Respuesta
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6 6 5 3 3 5 2 4 4 2 1 sin (1) 5*cos (1) cos (1)*sin(1) cos (1)*sin (1) sin (1)*cos(1) 3*cos (1)*sin (1) 13*cos (1)*sin (1) - --- + ------- + --------- - -------------- + --------------- + -------------- + ----------------- + ------------------ 288 48 144 16 6 16 32 96
$$- \frac{1}{288} - \frac{\sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{16} + \frac{5 \cos^{6}{\left(1 \right)}}{144} + \frac{\sin^{6}{\left(1 \right)}}{48} + \frac{13 \sin^{2}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{96} + \frac{3 \sin^{4}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} + \frac{\sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{16} + \frac{\sin^{3}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{6}$$
=
=
6 6 5 3 3 5 2 4 4 2 1 sin (1) 5*cos (1) cos (1)*sin(1) cos (1)*sin (1) sin (1)*cos(1) 3*cos (1)*sin (1) 13*cos (1)*sin (1) - --- + ------- + --------- - -------------- + --------------- + -------------- + ----------------- + ------------------ 288 48 144 16 6 16 32 96
$$- \frac{1}{288} - \frac{\sin{\left(1 \right)} \cos^{5}{\left(1 \right)}}{16} + \frac{5 \cos^{6}{\left(1 \right)}}{144} + \frac{\sin^{6}{\left(1 \right)}}{48} + \frac{13 \sin^{2}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{96} + \frac{3 \sin^{4}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{32} + \frac{\sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{16} + \frac{\sin^{3}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{6}$$
-1/288 + sin(1)^6/48 + 5*cos(1)^6/144 - cos(1)^5*sin(1)/16 + cos(1)^3*sin(1)^3/6 + sin(1)^5*cos(1)/16 + 3*cos(1)^2*sin(1)^4/32 + 13*cos(1)^4*sin(1)^2/96
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.