[size=200][b]7.-[/b][math]v\left(t\right)=\left\langle10,-32t+4\right\rangle,r\left(0\right)=\left\langle3,8\right\rangle[/math][/size][br][br]Integramos a v(t):[br][br] [math]r\left(t\right)=\int v\left(t\right)dt=\int\left[10i+\left(-32t+4\right)j\right]dt=10ti+\left(-16t^2+4t\right)j+c_1[/math][br][br]Ahora, buscamos las constantes de integración, que están en [math]c_1[/math], usando la condición inicial [math]r\left(0\right)=\left\langle3,8\right\rangle[/math]:[br][br] [math]\left\langle3,8\right\rangle=r\left(0\right)=\left(10\left(0\right)i+\left(-16\left(0\right)^2+4\left(0\right)\right)j\right)=\left\langle3,8\right\rangle[/math][br][br]Entonces, la función posición es:[br][br] [math]r\left(t\right)=\left\langle10t+3,-16t^2+4t+8\right\rangle[/math].

[size=200][b]9.-[/b][math]a\left(t\right)=\left\langle0,-32\right\rangle,v\left(0\right)=\left\langle5,0\right\rangle,r\left(0\right)=\left\langle0,16\right\rangle[/math].[br][br][/size]Integramos a [math]a\left(t\right)[/math]:[br][br] [math]v\left(t\right)=\int a\left(t\right)dt=\int\left[-32j\right]dt=-32tj+c_1[/math][br][br]Ahora, buscamos las constantes de integración, que están en [math]c_1[/math] usando la condición inicial [math]v\left(0\right)=\left\langle5,0\right\rangle[/math]:[br][br] [math]\left\langle5,0\right\rangle=v\left(0\right)=-32\left(0\right)j=\left\langle5,0\right\rangle[/math][br][br]Entonces, la función velocidad es:[br][br] [math]v\left(t\right)=\left\langle5,-32t\right\rangle[/math][br][br]Ahora, volvemos a integrar a [math]v\left(t\right)[/math]:[br][br] [math]r\left(t\right)=\int v\left(t\right)dt=\int\left[5i+\left(-32t\right)j\right]dt=\left(5t\right)i+\left(-16t^2\right)j+c_2[/math][br][br]Buscamos las constantes de integración, que están en [math]c_2[/math] usando la condición inicial [math]r\left(0\right)=\left\langle0,16\right\rangle[/math]:[br][br] [math]\left\langle0,16\right\rangle=r\left(0\right)=5\left(0\right)i+\left(-16\left(0\right)^2\right)j=\left\langle0,16\right\rangle[/math][br][br]Por lo tanto, la función posición es:[br][br] [math]r\left(t\right)=\left\langle5t,-16t^2+16\right\rangle[/math]

[size=200][b]11.-[/b][math]v\left(t\right)=\left\langle12\sqrt{t},\frac{t}{\left(t^2+1\right)},te^{-t}\right\rangle,r\left(0\right)=\left\langle8,-2,1\right\rangle[/math][br][br][/size]Integramos a [math]v\left(t\right)[/math]:[br][br] [math]r\left(t\right)=\int v\left(t\right)dt=\int\left[\left(12\sqrt{t}\right)i+\left(\frac{t}{\left(t^2+1\right)}\right)j+\left(te^{-t}\right)k\right]dt=\left(16t^{\frac{3}{2}}\right)i+\left(\frac{1}{2}ln\left(t^2+1\right)\right)j+\left(-te^{-t}-e^{-t}\right)k+c_1[/math][br][br]Entonces, buscamos las constantes de integración, que están en [math]c_1[/math] usando la condición inicial [math]r\left(0\right)=\left\langle8,-2,1\right\rangle[/math]:[br][br] [math]\left\langle8,-2,1\right\rangle=r\left(0\right)=\left(0\right)i+\left(0\right)j+\left(0\right)k=\left\langle8,-2,1\right\rangle[/math][br][br]Por lo tanto, la función posición es:[br][br] [math]r\left(t\right)=\left\langle16t^{\frac{3}{2}}+8,\frac{1}{2}ln\left(t^2+1\right)-2,-te^{-t}-e^{-t}+1\right\rangle[/math]

[size=200][b]13.-[/b][math]a\left(t\right)=\left\langle t,0,-16\right\rangle,v\left(0\right)=\left\langle12,-4,0\right\rangle,r\left(0\right)=\left\langle5,0,2\right\rangle[/math][br][/size][br]Integramos a [math]a\left(t\right)[/math]:[br][br] [math]v\left(t\right)=\int a\left(t\right)dt=\int\left[\left(t\right)i+\left(-16\right)k\right]dt=\frac{t^2}{2}i+\left(-16t\right)k+c_1[/math][br][br]Ahora, buscamos las constantes de integración que están en [math]c_1[/math] usando la condición inicial[br] [math]v\left(0\right)=\left\langle12,-4,0\right\rangle[/math]:[br][br] [math]\left\langle12,-4,0\right\rangle=v\left(0\right)=\left(0\right)i+\left(0\right)j+\left(0\right)k=\left\langle12,-4,0\right\rangle[/math][br][br]Entonces, la función velocidad es:[br][br] [math]v\left(t\right)=\left\langle\frac{t^2}{2}+12,-4,-16t\right\rangle[/math][br][br]Ahora, integramos a [math]v\left(t\right)[/math]:[br][br] [math]r\left(t\right)=\int v\left(t\right)dt=\int\left[\left(\frac{t^2}{2}+12\right)i+\left(-4\right)j+\left(-16t\right)k\right]dt=\left(\frac{1}{6}t^3+12t\right)i+\left(-4t\right)j+\left(-8t^2\right)k+c_2[/math][br][br]Entonces, buscamos las constantes de integración que están en [math]c_2[/math] usando la condición inicial [math]r\left(0\right)=\left\langle5,0,2\right\rangle[/math]:[br][br] [math]\left\langle5,0,2\right\rangle=r\left(0\right)=\left(0\right)i+\left(0\right)j+\left(0\right)k=\left\langle5,0,2\right\rangle[/math][br][br]Por lo tanto, la función posición es:[br][br] [math]r\left(t\right)=\left\langle\frac{1}{6}t^3+12t+5,-4t,-8t^2+2\right\rangle[/math][br]