Integrate a Function Using the Sine Case

By Mark Zegarelli

When the function you’re integrating includes a term of the form (a2bx2)n, draw your trig substitution triangle for the sine case. For example, suppose that you want to evaluate the following integral:

The integrating a function with a term being raised to a power.

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

One half

Here’s how you use trig substitution to handle the job:

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

    The trig substitution triangle.

    This figure shows you how to fill in the triangle for the sine case. Notice that the radical goes on the adjacent side of the triangle. Then, to fill in the other two sides of the triangle, you use the square roots of the two terms inside the radical — that is, 2 and x. Place 2 on the hypotenuse and x on the opposite side.

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

    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:

    Identifying the separate pieces of the integral that contain x.

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

    This is the real work of trig substitution, but when your triangle is set up properly, this work becomes a lot easier. In the sine case, all trig functions should be sines and cosines.

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

    The square root of four minus x squared.

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

    Representing the radical portion as a trig function of theta

    Because the numerator is the adjacent side of the triangle and the denominator is the hypotenuse

    A over H.

    this fraction is equal to

    Cosine of theta.

    Now a little algebra gets the radical alone on one side of the equation:

    getting the radical alone on one side of the equation

    Next, you want to express dx as a trig function of theta. To do so, build another fraction with the variable x in the numerator and the constant 2 in the denominator. Then set this fraction equal to the correct trig function:

    X divided by two equals the sine of theta.

    This time, the numerator is the opposite side of the triangle and the denominator is the hypotenuse

    The opposite side of the triangle divided by the hypothenuse.

    so this fraction is equal to

    Sine of Theta.

    Now solve for x and then differentiate:

    Solving the equation and differentiating.

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

    Rewriting the integral in terms of theta.

  5. To change those two theta terms into x terms, reuse the following equation:

    Changing the two theta terms into x.

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

    The substituted equation.

This answer is perfectly valid so, technically speaking, you can stop here. However, some professors frown upon the nesting of trig and inverse trig functions, so they’ll prefer a simplified version of

The answer to the problem.

To find this, start by applying the double-angle sine formula to

Simplifying the equation with the double-angle sine formula.

Now use your trig substitution triangle to substitute values for

Use the substitution triangle to substitute the values.

in terms of x:

The solution to a math problem in terms of x.

To finish, substitute this expression for that problematic second term to get your final answer in a simplified form:

The final answer in a simplified form