∫tan2xsecxdx

To solve the integral ∫tan^2(x)sec(x)dx, we can use the identity tan^2(x) = sec^2(x) - 1.

Step 1: Substitute tan^2(x) with sec^2(x) - 1. The integral becomes ∫(sec^2(x) - 1)sec(x)dx = ∫sec^3(x)dx - ∫sec(x)dx.

Step 2: Now, we have two separate integrals to solve. The integral of sec(x) is ln|sec(x) + tan(x)|.

Step 3: The integral of sec^3(x) is a bit more complicated. We can use the reduction formula: ∫sec^3(x)dx = 1/2 sec(x) tan(x) + 1/2 ln|sec(x) + tan(x)|.

Step 4: Putting it all together, the solution to the original integral is 1/2 sec(x) tan(x) + 1/2 ln|sec(x) + tan(x)| - ln|sec(x) + tan(x)|.

Step 5: Simplify the solution to get 1/2 sec(x) tan(x) - 1/2 ln|sec(x) + tan(x)|.

So, ∫tan^2(x)sec(x)dx = 1/2 sec(x) tan(x) - 1/2 ln|sec(x) + tan(x)| + C, where C is the constant of integration.