Sr Examen
Integral de ((sin^2)×x)×((cos^2)×x) dx
Solución
Ha introducido
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1 / | | 2 2 | sin (x)*x*cos (x)*x dx | / 0
$$\int\limits_{0}^{1} x \sin^{2}{\left(x \right)} x \cos^{2}{\left(x \right)}\, dx$$
Integral((sin(x)^2*x)*(cos(x)^2*x), (x, 0, 1))
Respuesta (Indefinida)
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/ | 4 4 3 3 4 3 4 3 2 3 2 3 3 2 2 2 2 | 2 2 x*cos (x) x*sin (x) sin (x)*cos(x) x *cos (x) x *sin (x) cos (x)*sin(x) x *cos (x)*sin(x) x *sin (x)*cos(x) x *cos (x)*sin (x) 3*x*cos (x)*sin (x) | sin (x)*x*cos (x)*x dx = C - --------- - --------- - -------------- + ---------- + ---------- + -------------- - ----------------- + ----------------- + ------------------ + ------------------- | 64 64 64 24 24 64 8 8 12 32 /
$$\int x \sin^{2}{\left(x \right)} x \cos^{2}{\left(x \right)}\, dx = C + \frac{x^{3} \sin^{4}{\left(x \right)}}{24} + \frac{x^{3} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{12} + \frac{x^{3} \cos^{4}{\left(x \right)}}{24} + \frac{x^{2} \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{8} - \frac{x^{2} \sin{\left(x \right)} \cos^{3}{\left(x \right)}}{8} - \frac{x \sin^{4}{\left(x \right)}}{64} + \frac{3 x \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{32} - \frac{x \cos^{4}{\left(x \right)}}{64} - \frac{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}{64} + \frac{\sin{\left(x \right)} \cos^{3}{\left(x \right)}}{64}$$
Gráfica
Respuesta
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4 4 3 3 2 2 5*cos (1) 5*sin (1) 7*cos (1)*sin(1) 7*sin (1)*cos(1) 17*cos (1)*sin (1) --------- + --------- - ---------------- + ---------------- + ------------------ 192 192 64 64 96
$$- \frac{7 \sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{64} + \frac{5 \cos^{4}{\left(1 \right)}}{192} + \frac{5 \sin^{4}{\left(1 \right)}}{192} + \frac{7 \sin^{3}{\left(1 \right)} \cos{\left(1 \right)}}{64} + \frac{17 \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{96}$$
=
=
4 4 3 3 2 2 5*cos (1) 5*sin (1) 7*cos (1)*sin(1) 7*sin (1)*cos(1) 17*cos (1)*sin (1) --------- + --------- - ---------------- + ---------------- + ------------------ 192 192 64 64 96
$$- \frac{7 \sin{\left(1 \right)} \cos^{3}{\left(1 \right)}}{64} + \frac{5 \cos^{4}{\left(1 \right)}}{192} + \frac{5 \sin^{4}{\left(1 \right)}}{192} + \frac{7 \sin^{3}{\left(1 \right)} \cos{\left(1 \right)}}{64} + \frac{17 \sin^{2}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{96}$$
5*cos(1)^4/192 + 5*sin(1)^4/192 - 7*cos(1)^3*sin(1)/64 + 7*sin(1)^3*cos(1)/64 + 17*cos(1)^2*sin(1)^2/96
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.