How to integrate sec^2(x)tan^2(x)?

To integrate sec^2(x)tan^2(x), use the substitution u = tan(x).

Integrating sec^2(x)tan^2(x) can be done using the substitution u = tan(x). This substitution allows us to express sec^2(x) in terms of u, as sec^2(x) = 1 + tan^2(x) = 1 + u^2. Similarly, we can express tan^2(x) in terms of u, as tan^2(x) = u^2.

Substituting these expressions into the integral, we get:

∫sec^2(x)tan^2(x) dx = ∫(1 + u^2)(u^2) du

Expanding the integrand, we get:

∫(1 + u^2)(u^2) du = ∫(u^2 + u^4) du

Integrating each term separately, we get:

∫u^2 du + ∫u^4 du = (u^3)/3 + (u^5)/5 + C

Substituting back u = tan(x), we get the final answer:

∫sec^2(x)tan^2(x) dx = (tan^3(x))/3 + (tan^5(x))/5 + C