How Do You Solve for the Integral of Sec(x)?

By Staff WriterLast Updated Apr 16, 2020 8:14:16 PM ET

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.