To find the antiderivative of tan²(x) dx, we can use a trigonometric identity to simplify our work. The identity we will use is:
tan²(x) = sec²(x) – 1
This means that we can rewrite the integral as:
∫ tan²(x) dx = ∫ (sec²(x) – 1) dx
This can be split into two separate integrals:
∫ tan²(x) dx = ∫ sec²(x) dx – ∫ 1 dx
We know the integral of sec²(x) is:
∫ sec²(x) dx = tan(x)
And the integral of 1 is simply:
∫ 1 dx = x
Putting this together, we have:
∫ tan²(x) dx = tan(x) – x + C
where C is the constant of integration.
Therefore, the antiderivative of tan²(x) dx is:
tan(x) – x + C