Trig substitution with completing the square: integral of 1/(x^2-4x) using a secant substitution.

In this problem, we use a secant substitution to calculate the indefinite integral of 1/(x^2-4x).

Note that this integral is most naturally done using partial fractions decomposition, but we already did it that way over here:    • Integrate 1/(x^2-4x):  partial fracti...  

In this trig substitution integral, we have to start by completing the square in order to fit the form (variable)^2-constant. So we begin by completing the square on x^2-4x to get (x-2)^2-4.

Next, we show how to figure out which trig substitution to use: we start with the secant/tangent trig identity and solve for tan^2(theta) as sec^2(theta)-1, which fits the form of our denominator.

Now we can make the trigonometric substitution x-2=2sec(theta) and transform the integral to theta space. We transform the integral in terms of theta and take advantage of our trig identity to clean things up, then we're left with the integral of cosecant(theta), which integrates to ln|csc(theta)+cot(theta)|.

Now we have to replace theta in terms of x, and that requires solving for theta in our original trig substitution, to get arcsec((x-2)/2). So we find ourselves having to calculate the cosecant of an inverse secant and the cotangent of an inverse secant, and for this we use a right triangle to visualize the angle whose secant is x-2 over 2.

Once we simplify the trig functions of inverse trig functions as algebraic expressions in terms of x, we can apply log properties to simplify the answer and compare to the result given by the partial fractions approach.