Integrate the function.
\[
\int 5 \tan ^{2} x \sec ^{2} x d x
\]

Step 1 ：Given the integral \(\int 5 \tan ^{2} x \sec ^{2} x d x\)

Step 2 ：We can simplify the integral by using the identity \(\tan^2(x) = \sec^2(x) - 1\). This simplifies the integral to \(5*(\sec^2(x))^2 - 5*\sec^2(x) dx\)

Step 3 ：We can then integrate each term separately. The integral of \(\sec^2(x) dx\) is \(\tan(x)\), and the integral of \((\sec^2(x))^2 dx\) can be found using the power rule for integration

Step 4 ：Applying these rules, we find that the integral of the function \(5*\tan^2(x)*\sec^2(x) dx\) is \(-5*\sin(x)/(3*\cos(x)) + 5*\sin(x)/(3*\cos(x)^3)\)

Step 5 ：Final Answer: \(\boxed{-\frac{5 \sin x}{3 \cos x} + \frac{5 \sin x}{3 \cos ^{3} x}}\)