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