Sr Examen
Integral de x*sinx*cos(2x/pi) dx
Solución
Ha introducido
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pi / | | /2*x\ | x*sin(x)*cos|---| dx | \ pi/ | / 0
$$\int\limits_{0}^{\pi} x \sin{\left(x \right)} \cos{\left(\frac{2 x}{\pi} \right)}\, dx$$
Integral((x*sin(x))*cos((2*x)/pi), (x, 0, pi))
Respuesta (Indefinida)
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/ 4 / 2*x\ 4 / 2*x\ 3 / 2*x\ 2 / 2*x\ 2 / 2*x\ 3 / 2*x\ 4 / 2*x\ 4 / 2*x\ / 2*x\ 3 / 2*x\ 3 / 2*x\ 2 / 2*x\ 2 / 2*x\ / 2*x\ | pi *sin|x - ---| pi *sin|x + ---| 4*pi *sin|x + ---| 4*pi *sin|x - ---| 4*pi *sin|x + ---| 4*pi *sin|x - ---| x*pi *cos|x - ---| x*pi *cos|x + ---| 8*pi*x*cos|x + ---| 2*x*pi *cos|x - ---| 2*x*pi *cos|x + ---| 4*x*pi *cos|x - ---| 4*x*pi *cos|x + ---| 8*pi*x*cos|x - ---| | /2*x\ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ \ pi/ | x*sin(x)*cos|---| dx = C + ------------------- + ------------------- - ------------------- + ------------------- + ------------------- + ------------------- - ------------------- - ------------------- - ------------------- - -------------------- + -------------------- + -------------------- + -------------------- + ------------------- | \ pi/ 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 | 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi 32 - 16*pi + 2*pi /
$$\int x \sin{\left(x \right)} \cos{\left(\frac{2 x}{\pi} \right)}\, dx = C - \frac{\pi^{4} x \cos{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} - \frac{2 \pi^{3} x \cos{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{8 \pi x \cos{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{2} x \cos{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} - \frac{\pi^{4} x \cos{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} - \frac{8 \pi x \cos{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{2} x \cos{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{2 \pi^{3} x \cos{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{2} \sin{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{\pi^{4} \sin{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{3} \sin{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} - \frac{4 \pi^{3} \sin{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{2} \sin{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{\pi^{4} \sin{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}}$$
Gráfica
Respuesta
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5 3 3
pi *cos(2) 4*pi *cos(2) 4*pi *sin(2)
---------------- - ---------------- + ----------------
4 2 4 2 4 2
16 + pi - 8*pi 16 + pi - 8*pi 16 + pi - 8*pi
$$\frac{\pi^{5} \cos{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} - \frac{4 \pi^{3} \cos{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} + \frac{4 \pi^{3} \sin{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}}$$
=
=
5 3 3
pi *cos(2) 4*pi *cos(2) 4*pi *sin(2)
---------------- - ---------------- + ----------------
4 2 4 2 4 2
16 + pi - 8*pi 16 + pi - 8*pi 16 + pi - 8*pi
$$\frac{\pi^{5} \cos{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} - \frac{4 \pi^{3} \cos{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} + \frac{4 \pi^{3} \sin{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}}$$
pi^5*cos(2)/(16 + pi^4 - 8*pi^2) - 4*pi^3*cos(2)/(16 + pi^4 - 8*pi^2) + 4*pi^3*sin(2)/(16 + pi^4 - 8*pi^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.