how do i evaluate the integral of secxdx
To evaluate the integral of sec(x) dx, you can use a technique called substitution. Here's a step-by-step explanation on how to do it:
Step 1: Start with the integral of sec(x) dx:
∫ sec(x) dx
Step 2: Identify a substitution. In this case, let's substitute u = tan(x). To find du, take the derivative of both sides with respect to x:
du/dx = sec^2(x)
dx = du / sec^2(x)
Step 3: Substitute the values in terms of u and du into the integral:
∫ sec(x) dx = ∫ (1/ sec(x)) (du / sec^2(x))
Step 4: Simplify the expression:
∫ (1/ sec(x)) (du / sec^2(x)) = ∫ (1/ sec(x)) (du / sec^2(x)) = ∫ (du / sec(x))
= ∫ du
Step 5: Evaluate the integral:
∫ du = u + C
Step 6: Substitute back the original variable:
u + C = tan(x) + C
So, the final answer for ∫ sec(x) dx is tan(x) + C, where C represents the constant of integration.