Evaluate the integral. \[ \int \frac{4}{\sqrt{x}(1+4 \sqrt{x})^{7}} d x \] \[ \int \frac{4}{\sqrt{x}(1+4 \sqrt{x})^{7}} d x= \]

Step 1 ：First, we can make a substitution to simplify the integral. Let \(u = 1 + 4\sqrt{x}\). Then, \(du = 2dx/\sqrt{x}\).

Step 2 ：Substituting these into the integral, we get \(\int \frac{2}{u^{7}} du\).

Step 3 ：This is a standard form of integral. The antiderivative of \(u^{-n}\) is \(-\frac{1}{(n-1)u^{n-1}}\), for \(n \neq 1\).

Step 4 ：So, the antiderivative of \(2u^{-7}\) is \(-\frac{2}{6u^{6}} = -\frac{1}{3u^{6}}\).

Step 5 ：Substituting back for \(u\), we get \(-\frac{1}{3(1 + 4\sqrt{x})^{6}}\).

Step 6 ：So, the integral of \(\frac{4}{\sqrt{x}(1+4 \sqrt{x})^{7}} dx\) is \(-\frac{1}{3(1 + 4\sqrt{x})^{6}} + C\), where \(C\) is the constant of integration.

Step 7 ：Finally, we check that this result satisfies the requirements of the problem. It is in simplest form, and it is the antiderivative of the original integrand, so it is the correct solution.