Compute $\int_{0}^{\ln (2) / 4} \frac{e^{4 x}}{\left(e^{4 x}+4\right)^{7}} d x$

Step 1 ：Let \( u = e^{4x} + 4 \), then \( du = 4e^{4x}dx \)

Step 2 ：Change the limits of integration to match the substitution: lower limit \( u(0) = 5 \), upper limit \( u(\ln(2)/4) = 2^{1/4} + 4 \)

Step 3 ：Rewrite the integral in terms of \( u \) and \( du \): \( \int \frac{1}{4u^{7}} du \)

Step 4 ：Integrate to find \( -\frac{5}{4u^{6}} \) evaluated from \( 5 \) to \( 2^{1/4} + 4 \)

Step 5 ：Plug in the limits of integration: \( -\frac{5}{4(2^{1/4} + 4)^{6}} + \frac{5}{4(5)^{6}} \)

Step 6 ：Simplify the expression

Step 7 ：\( \boxed{-\frac{5}{4 \left(e^{4 x}+4\right)^{6}}+\frac{2^{1 / 4}+4}{4 \left(e^{4 x}+4\right)^{6}}} \)