Solving the integral of sec(x) tan(x) using the substitution method | step-by-step guide

S sec x tan x dx =

To solve the integral ∫ sec(x) tan(x) dx, we can use the substitution method

To solve the integral ∫ sec(x) tan(x) dx, we can use the substitution method.

Let’s start by using the identity:
sec(x) = 1/cos(x)

Now, let u = cos(x).
Taking the derivative of u with respect to x, we get du = -sin(x) dx, or -du = sin(x) dx.

Substituting these values into our integral, we have:
∫ sec(x) tan(x) dx = ∫ (1/cos(x)) * (sin(x)/cos(x)) dx

Using the substitution u = cos(x) and -du = sin(x) dx, we can rewrite the integral as:
∫ (1/u) * (-du/u)

Simplifying:
-∫ du/u^2

Integrating, we get:
-(-1/u) = 1/u

Since we substituted u = cos(x), we can substitute back to get:
∫ sec(x) tan(x) dx = 1/cos(x) = sec(x) + C

So, the final solution is:
∫ sec(x) tan(x) dx = sec(x) + C, where C is the constant of integration.

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