# Integrate a Function Using the Secant Case

When the function that you’re integrating includes a term of the form (*bx*^{2}* *– *a*^{2})^{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 *x*^{2} minus a constant is being raised to a power

Integrate using trig substitution as follows:

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*x**.*Place the constant 1 on the adjacent side and the variable 4*x*on the hypotenuse.You can check to make sure that this placement is correct by using the Pythagorean theorem:

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**:*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 radicalas 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**:*Express the integral in terms of theta and evaluate it:

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

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: