∫ sec^2 x
tan x
We can approach this problem in a couple of ways, but one common method is to use the trigonometric identity:
sec^2 x = 1 + tan^2 x
To see why this identity is true, recall that:
sec^2 x = (1/cos x)^2 = 1/cos^2 x
Then, using the identity cos^2 x + sin^2 x = 1, we have:
1/cos^2 x = (cos^2 x + sin^2 x)/cos^2 x = 1 + sin^2 x/cos^2 x
But sin^2 x/cos^2 x = tan^2 x, so we obtain:
sec^2 x = 1 + tan^2 x
Now we can use this identity to evaluate the integral:
∫ sec^2 x dx = ∫ (1 + tan^2 x) dx
= ∫ 1 dx + ∫ tan^2 x dx
= x + ∫ (sec^2 x – 1) dx (using the substitution u = tan x)
= x + tan x – x + C (where C is the constant of integration)
= tan x + C
Therefore, the antiderivative of sec^2 x is tan x + C.
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