Final answer:
The integral of tan²(x) can be determined by applying the identity tan²(x) = sec²(x) - 1, then integrating the resulting functions to get tan(x) - x + C, where C is the constant of integration.
Step-by-step explanation:
Integral of tan²(x)
To find the integral of tan²(x), we can use a trigonometric identity to rewrite the expression in a more easily integrable form. We start with the identity tan²(x) = sec²(x) - 1, which comes from the Pythagorean identity for trigonometric functions:
tan²(x) + 1 = sec²(x).
This allows us to express the integral as
∫ tan²(x) dx = ∫ (sec²(x) - 1) dx
The integral can now be separated into two parts:
∫ (sec²(x) - 1) dx = ∫ sec²(x) dx - ∫ dx
The integral of sec²(x) is simply tan(x), and the integral of 1 with respect to x is x. Therefore, the integral of tan²(x) reduces to:
∫ tan²(x) dx = tan(x) - x + C,
where C represents the constant of integration. Note: It’s important to remember that when integrating trigonometric functions, we must ensure that the function is integrable over the interval of interest, and consider whether additional techniques like substitution might be required for more complex scenarios.