[tex]a(t)=10\cos5t[/tex]
[tex]v(t)=\displaystyle\int 10\cos5t\,\mathrm dt=2\sin5t+C_1[/tex]
Since [tex]v(0)=-10[/tex], you have
[tex]-10=2\sin0+C_1\implies C_1=-10[/tex]
so
[tex]v(t)=2\sin5t-10[/tex]
Next,
[tex]s(t)=\displaystyle\int(2\sin5t-10)\,\mathrm dt=-\dfrac25\cos5t-10t+C_2[/tex]
Since [tex]s(0)=10[/tex], you get
[tex]10=-\dfrac25\cos0-10\times0+C_2\implies C_2=\dfrac{52}5[/tex]
and so
[tex]s(t)=-\dfrac25\cos5t-10t+\dfrac{52}5[/tex]
