How do I integrate tan^2 x?

Firstly, use the trigonometric formula tan²x = sec²x - 1, which you can easily derive from sin²x + cos²x =1, by dividing both sides by cos²x and re-arranging. Now, you should remember that differentiating tan x gives sec²x. Therefore, the opposite is true for integration, integrating sec²x gives tan x. Also, differentiating x gives 1, hence, integrating 1 gives x. With this knowledge you can express tan²x as sec²x - 1 and integrate it to give tan x -x +C.