How To Integrate Sec(X)Tan(X) Using U-Substitution

∫secxtanx dx =

secx + C

To integrate ∫secxtanx dx, we can use u-substitution. Let u = secx + tanx, then du/dx = secxtanx + sec^2x.

Now, we need to manipulate the integral so that it involves du/dx and dx only. We can rewrite secxtanx as (secx)(tanx), and use the identity tan^2x + 1 = sec^2x, to obtain:

∫(secx)(tanx) dx = ∫[(tan^2x + 1)/(secx)] dx

Substituting u = secx + tanx, we have:

∫[(tan^2x + 1)/(secx)] dx = ∫[(u^2 – 1)/u] (du/dx) dx

Notice that we can arrange the left side of the equation into (-1/u)(-u^2 + 1). Thus, we have:

∫(-1/u)(-u^2 + 1) (du/dx) dx

Integrating, we get:

∫(-1/u)(-u^2 + 1) (du/dx) dx = ∫(-1/u) du = -ln|u| + C

Substituting back in our expression for u, we have:

∫secxtanx dx = -ln|secx + tanx| + C

Therefore, the antiderivative of secxtanx is -ln|secx + tanx| + C.

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