How to Solve the Integral of Sec(x) Using Substitution Method

∫secx dx

ln|secx + tanx| + C

We can solve this integral using the substitution method. Let’s start by selecting u = tan(x) and then finding an expression for dx in terms of u:

u = tan(x)
du/dx = sec^2(x)
dx = du / sec^2(x)

Substituting this into the integral, we get:

∫secx dx = ∫secx (du / sec^2(x)) = ∫du / cos(x)

Now, we need to express the integrand in terms of u, so that we can integrate with respect to u:

cos(x) = 1 / sqrt(1 + tan^2(x)) = 1 / sqrt(1 + u^2)

Substituting this expression back into the integral, we get:

∫du / cos(x) = ∫du / (1 / sqrt(1 + u^2)) = ∫sqrt(1 + u^2) du

This integral can be solved using a simple substitution:

Let u = sinh(v), then du = cosh(v) dv

Substituting this into the integral, we get:

∫sqrt(1 + u^2) du = ∫cosh(v) * sqrt(1 + sinh^2(v)) dv
= ∫cosh^2(v) dv

Now we use the identity cosh^2(v) = (1/2)(cosh(2v) + 1) to solve the integral:

∫cosh^2(v) dv = (1/2)∫(cosh(2v) + 1) dv
= (1/2)(sinh(2v)/2 + v) + C
= (1/4)(e^(2x) – 1) + (1/2)arctan(u) + C

Substituting back for u, we get:

∫sec(x) dx = (1/4)(e^(2x) – 1) + (1/2)arctan(tan(x)) + C

Therefore, the integral of sec(x) is (1/4)(e^(2x) – 1) + (1/2)arctan(tan(x)) + C, where C is an arbitrary constant of integration.

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