∫sec^2x dx =
tanx + C
The integral of sec^2x can be calculated using integration by substitution method. Let u = tan x, then du/dx = sec^2x and dx = du/sec^2x. Substituting these values, we get:
∫sec^2x dx = ∫1/(cos^2x) dx
Let u = tan x, then cos^2x = 1/(1+tan^2x) = 1/(1+u^2)
Using this substitution, we obtain:
∫1/(cos^2x) dx = ∫(1+u^2) du
= u + (1/3)u^3 + C
Substituting back u = tan x, we get:
∫sec^2x dx = tan x + (1/3)tan^3 x + C
Therefore, the antiderivative of sec^2x is tan x + (1/3)tan^3 x + C.
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