How can I solve for the anti-derivative of 1 / (secsquared x) dx ?

Answer 1

Integral (1 / sec^2(x) ) dx

The first thing to notice that is secant is just the reciprocal of cosine. For that reason, 1 over secant is the same as 1 over 1 over cosine, so our integral becomes

Integral (cos^2(x) ) dx

From here, you use the half angle identity. Recall that

cos^2(x) = (1/2) (1 + cos2x). Therefore, we have

Integral ( (1/2) (1 + cos2x) ) dx

Pull the constant (1/2) out of the integral, and we get

(1/2) * Integral (1 + cos2x) dx

Now, integrate.

(1/2) * [x + (1/2)sin(2x)] + C

(1/2)x + (1/4)sin(2x) + C

Answer 2

Integrate 1/sec²x with respect to x.

First rearrange the terms to make it easier to integrate.

1/sec²x = cos²x = [1 + cos(2x)]/2

Now we can integrate.

∫dx/sec²x = ∫{[1 + cos(2x)]/2}dx = x/2 + sin(2x)/4 + C
= x/2 + (sinx)(cosx)/2 + C = (1/2){x + (sinx)(cosx)} + C