The rate of expenditure for maintenance of a particular machine is given by $M^{\prime}(x)=9 x \sqrt{x^{2}+5}$, where $\mathrm{x}$ is time measured in years. Total maintenance costs through the second year are $\$ 119$. Find the total maintenance function by integration.

Step 1 ：Given the rate of expenditure for maintenance of a machine is \(M'(x)=9x\sqrt{x^{2}+5}\), where \(x\) is time measured in years.

Step 2 ：We need to find the total maintenance function, \(M(x)\), by integrating the rate of expenditure function, \(M'(x)\).

Step 3 ：Integrating \(M'(x)\) gives us \(M(x) = 3x^{2}\sqrt{x^{2}+5} + 15\sqrt{x^{2}+5} + C\), where \(C\) is the constant of integration.

Step 4 ：We know that the total maintenance costs through the second year are $119. We can use this information to find the value of \(C\).

Step 5 ：Setting \(M(2) = 119\), we get \(C + 81 = 119\). Solving this equation gives us \(C = 38\).

Step 6 ：Substituting \(C = 38\) into our equation for \(M(x)\), we get \(M(x) = 3x^{2}\sqrt{x^{2}+5} + 15\sqrt{x^{2}+5} + 38\).

Step 7 ：\(\boxed{M(x) = 3x^{2}\sqrt{x^{2}+5} + 15\sqrt{x^{2}+5} + 38}\) is the total maintenance function.