91Ó°ÊÓ

To solve definite integrals, it can be helpful to simplify the integrand. The integrand is the part of the integral that we are integrating. A simpler integrand makes it easier to find the antiderivative.

In this exercise, we start with the integrand \(8x(x+4)^{-3}\). Attempting to simplify by factoring or simplifying algebraically might not always be straightforward.

By closely looking at this specific integral, we find that splitting further or factoring parts don't simplify it greatly. Therefore, sometimes the form is already the most simplified it can become for practical purposes of integration.

When faced with a complex integrand, the substitution method can be very effective. This method simplifies the integration process by changing variables.

For the given definite integral \(\begin{aligned}\tag{1}\int_{0}^{4} 8 x(x+4)^{-3} \, d x,\tag{2}\end{aligned}\) we use substitution.

We let \(u = x+4\). This means our new differential becomes \(du = dx\). When we change variables from \(x\) to \(u\), we also need to update our limits of integration.

When \(x=0\), then \(u=4\). When \(x=4\), then \(u=8\). Now the integral can be rewritten as: \(\begin{aligned}\tag{3}\int_{4}^{8} 8 (u-4) u^{-3} \, d u,\tag{4}\end{aligned}\) simplifying the process by dealing with a new and often simpler integrand.

These new boundaries and integrand simplification will help in easily computing the integral.

The limits of integration define the range over which the integral is computed. When using the substitution method, updating the limits of integration to match the new variable is critical.

Originally, the limits for \(x\) were from 0 to 4. After the substitution \(u = x+4\), we need to recalculate limits in terms of \(u\).

At \(x = 0\), \(u = 4\). At \(x = 4\), \(u = 8\). So, the integral \(\begin{aligned}\tag{5}\int_{0}^{4} 8x (x+4)^{-3}\tag{6}\, d x,\end{aligned}\) transforms to \(u\) limits of 4 to 8.

Hence, our new integral becomes more manageable:

\(\begin{aligned}\tag{7}\int_{4}^{8} 8 u^{-2} \, d u,\tag{8}\end{aligned}\) ensuring accurate calculation within the defined range.