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.
- 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.
- Set the value for u
Set the value of u as equal to sec(x) + tan(x).
- 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.
- 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)).
- Integrate du/u
Using basic integration rules, determine the integral of du/u as ln |u| + C.
- 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.