How do you solve for the integral of sec(x)?

Solve the integral of sec(x) by using the integration technique known as substitution. The technique is derived from the chain rule used in differentiation. The problem requires a knowledge of calculus and the trigonometric identities for differentiation.

1. Multiply sec(x) by a value equal to one

   In the integral, multiply sec(x) by (sec(x) + tan(x))/(sec(x) + tan(x)). Since the value is the same in both the numerator and denominator, it is equivalent to multiplying by one, which leaves the original value unchanged.

2. Set the value for u

   Set the value of u as equal to sec(x) + tan(x).

3. Find the value of du

   Based on the standard differential properties of trigonometric equations, du, the derivative of u, is set to (sec(x) * tan(x) + sec2(x)) dx.

4. Substitute in for u and du

   Substitute u and du into the given equation to get du/u. Substitute into sec(x) by (sec(x) + tan(x))/(sec(x) + tan(x)) to get the equation (sec(x) * tan(x) + sec2(x))/(sec(x) * tan(x)).

5. Integrate du/u

   Using basic integration rules, determine the integral of du/u as ln |u| + C.

6. Substitute out for u

   In the equation ln |u| + C, replace the u with its given value of sec(x) + tan(x) to get the final solution of ln |sec(x) + tan(x)| + C.