Learn How To Find The Antiderivative Of Sec^2 X Using A Trigonometric Identity

∫ 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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