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:

image0.png

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

image1.png

Integrate using trig substitution as follows:

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

    image2.jpg

    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:

    image3.png
  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:

    image4.png
  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

    image5.png

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

    image6.png

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

    image7.png

    so it equals

    image8.png

    Simplifying it a bit gives you this equation:

    image9.png

    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:

    image10.png

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

    image11.png

    which equals

    image12.png

    Now solve for x and differentiate to find dx:

    image13.png
  4. Express the integral in terms of theta and evaluate it:

    image14.png

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

    image15.png
  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

    image16.png

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

    image17.png
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