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