The solution of the differential equation $du=(16+u^2)dx$ is:

Given the differential equation:
$$du=(16+u^2)dx$$
Rearranging the terms to separate the variables:
$$\frac{du}{16+u^2}=dx$$
Integrating both sides:
$$\int \frac{du}{4^2+u^2}=\int dx$$
Using the standard integral formula $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)$:
$$\frac{1}{4}{\tan }^{-1}\left(\frac{u}{4}\right)=x+c$$
Multiplying by 4:
$${\tan }^{-1}\left(\frac{u}{4}\right)=4x+4c$$
Let $4c=C$:
$$\frac{u}{4}=\tan (4x+C)$$
$$u=4\tan (4x+C)$$