Integrate a Function Using the Tangent Case

By Mark Zegarelli

When the function you’re integrating includes a term of the form (a2 + x2)n, draw your trigonometry substitution triangle for the tangent case. For example, suppose that you want to evaluate the following integral:

  • Draw the trig substitution triangle for the tangent case.

    Drawing the trig substitution triangle for the tangent case

    The figure shows you how to fill in the triangle for the tangent case. Notice that the radical of what’s inside the parentheses goes on the hypotenuse 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, 2 and 3x. Place the constant term 2 on the adjacent side and the variable term 3x on the opposite side.

    With the tangent case, make sure not to mix up your placement of the variable and the constant.

  • 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:

    The two separate pieces of an integral that contain x

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

    In the tangent case, all trig functions should be initially expressed as tangents and secants.

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

    The square root of four plus nine times x squared.

    as the numerator and the constant 2 as the denominator. Then set this fraction equal to the appropriate trig function:

    Setting a fraction equal to secant of theta.

    Because this fraction is the hypotenuse of the triangle over the adjacent side

    the hypotenuse of the triangle over the adjacent side.

    it’s equal to

    Secant of theta.

    Now use algebra and trig identities to tweak this equation into shape:

    Using algebra and trig identities to tweak the equation.

    Next, express dx as a trig function of theta. To do so, build another fraction with the variable 3x in the numerator and the constant 2 in the denominator:

    3x/2 = tangent of theta.

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

    he opposite side of the triangle over the adjacent side

    so it equals

    Tangent of theta.

    Now solve for x and then differentiate:

    Solve the equation for x and differentiate.

  • Express the integral in terms of theta and evaluate it:

    Express the integral in terms of theta and evaluate it

    Now some cancellation and reorganization turns this nasty-looking integral into something manageable:

    A manageable integral.

    At this point, you can evaluate this integral:

    Evaluating a trigonometric integral.

    So here’s the substitution:

    The substitution for an integral

    And here is the antiderivative:

    The antiderative for a function.

  • Change the two theta terms back into x terms:

    You need to find a way to express theta in terms of x. Here’s the simplest way:

    Changing the theta terms in an equation into x terms.

    So here’s a substitution that gives you an answer:

    The solution to the substitution of the theta terms in a function.

  • This answer is valid, but most professors won’t be crazy about that ugly second term, with the sine of an arctangent. To simplify it, apply the double-angle sine formula to

    applying the double-angle sine formula to an equation.

    Now use your trig substitution triangle to substitute values for

    Sine and cosine.

    in terms of x:

    An equation expressed in terms of x.

    Finally, use this result to express the answer in terms of x:

    The solution to the integral of a function.