Integrate tan^2x

To integrate tan^2x, also written as ∫tan²x dx, tan squared x, and (tan x)^2, we start by using standard trig identities to simplify the integral.

We start by using the Pythagorean trig identity sin²x+cos²x=1.

We divide throughout by cos²x. The reason for this will become apparent in the following step.

We can now use this expression to substitute into the previous one.

Here we have an expression with tan²x, and all is moving according to plan.

We rearrange the expression for tan²x by moving the 1 to the other side.

As you can see, we now have a simpler integration that means the same thing. Although the second term, which is a constant, is simple to integrate, we need to focus our attention on the first term.

We start by using a well-known standard trig identity as shown above.

If we were to square both sides then this is the result.

Hence integration of the first term is equivalent to this new version which means the same thing.

Experience of differentiation reminds us that the differential of tan x is sec²x. This is usually found in standard formula booklets used in exams.

Therefore, we now have an integral for sec²x.

Hence, the integral for the first term is shown above.

We remind ourselves of the expression we previously found, and can now substitute the answer for the first term.

Hence, this is the solution, where C is the integration constant.