Sr Examen
Integral de x*sinx*cos(x/pi) dx
Solución
Ha introducido
[src]
pi / | | /x \ | x*sin(x)*cos|--| dx | \pi/ | / 0
$$\int\limits_{0}^{\pi} x \sin{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}\, dx$$
Integral((x*sin(x))*cos(x/pi), (x, 0, pi))
Respuesta (Indefinida)
[src]
/ 2 /x \ 4 /x \ 3 /x \ /x \ 2 /x \ 3 /x \ 4 /x \ | pi *cos|--|*sin(x) pi *cos|--|*sin(x) 2*pi *cos(x)*sin|--| pi*x*sin(x)*sin|--| x*pi *cos(x)*cos|--| x*pi *sin(x)*sin|--| x*pi *cos(x)*cos|--| | /x \ \pi/ \pi/ \pi/ \pi/ \pi/ \pi/ \pi/ | x*sin(x)*cos|--| dx = C + ------------------ + ------------------ - -------------------- + ------------------- + -------------------- - -------------------- - -------------------- | \pi/ 4 2 4 2 4 2 4 2 4 2 4 2 4 2 | 1 + pi - 2*pi 1 + pi - 2*pi 1 + pi - 2*pi 1 + pi - 2*pi 1 + pi - 2*pi 1 + pi - 2*pi 1 + pi - 2*pi /
$$\int x \sin{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}\, dx = C - \frac{\pi^{3} x \sin{\left(x \right)} \sin{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi x \sin{\left(x \right)} \sin{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} - \frac{\pi^{4} x \cos{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{2} x \cos{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{2} \sin{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{4} \sin{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} - \frac{2 \pi^{3} \sin{\left(\frac{x}{\pi} \right)} \cos{\left(x \right)}}{- 2 \pi^{2} + 1 + \pi^{4}}$$
Gráfica
Respuesta
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5 3 3
pi *cos(1) pi *cos(1) 2*pi *sin(1)
--------------- - --------------- + ---------------
4 2 4 2 4 2
1 + pi - 2*pi 1 + pi - 2*pi 1 + pi - 2*pi
$$- \frac{\pi^{3} \cos{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{2 \pi^{3} \sin{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{5} \cos{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}}$$
=
=
5 3 3
pi *cos(1) pi *cos(1) 2*pi *sin(1)
--------------- - --------------- + ---------------
4 2 4 2 4 2
1 + pi - 2*pi 1 + pi - 2*pi 1 + pi - 2*pi
$$- \frac{\pi^{3} \cos{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{2 \pi^{3} \sin{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{5} \cos{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}}$$
pi^5*cos(1)/(1 + pi^4 - 2*pi^2) - pi^3*cos(1)/(1 + pi^4 - 2*pi^2) + 2*pi^3*sin(1)/(1 + pi^4 - 2*pi^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.