Integrate a Function Using the Secant Case

When the function that you’re integrating includes a term of the form (bx² – a²)ⁿ, draw your trig substitution triangle for the secant case. For example, suppose that you want to evaluate this integral:

This is a secant case, because a multiple of x² minus a constant is being raised to a power

Integrate using trig substitution as follows:

1. Draw the trig substitution triangle for the secant case.

   

   The figure shows you how to fill in the triangle for the secant case. Notice that the radical goes on the opposite side of the triangle. Then, to fill in the other two sides of the triangle, use the square roots of the two terms inside the radical — that is, 1 and 4x. Place the constant 1 on the adjacent side and the variable 4x on the hypotenuse.

   You can check to make sure that this placement is correct by using the Pythagorean theorem:

   

2. Identify the separate pieces of the integral (including dx) that you need to express in terms of theta.

   In this case, the function contains two separate pieces that contain x:

   

3. Express these pieces in terms of trig functions of theta.

   In the secant case, all trig functions should be initially represented as tangents and secants.

   To represent the radical portion as a trig function of theta, build a fraction by using the radical

   

   as the numerator, and the constant 1 as the denominator. Then set this fraction equal to the appropriate trig function:

   

   Notice that this fraction is the opposite side of the triangle over the adjacent side

   

   so it equals

   

   Simplifying it a bit gives you this equation:

   

   Next, express dx as a trig function of theta. To do so, build another fraction with the variable x in the numerator and the constant 1 in the denominator:

   

   This time, the fraction is the hypotenuse over the adjacent side of the triangle

   

   which equals

   

   Now solve for x and differentiate to find dx:

   

4. Express the integral in terms of theta and evaluate it:

   

   Now use the formula for the integral of the secant function:

   

5. Change the two theta terms back into x terms:

   In this case, you don’t have to find the value of theta because you already know the values of

   

   in terms of x from Step 3. So substitute these two values to get your final answer: