Integrals 215 problem 06.mws

1. Integrating functions which involve quadratic expressions of the variable.

Integrals involving quadratic expressions of the form a*x^2+b*x+c are often evaluated with the help of completing the square to convert it to a form that looks like either r^2+u^2 , r^2-u^2 or u^2-r^2 . These expressions can often be evaluated directly with the help of an integral table. Maple can also evaluate these integrals as follows:

> Int(1/sqrt(r^2-u^2),u)=int(1/sqrt(r^2-u^2),u);

Int(1/(sqrt(r^2-u^2)),u) = arctan(u/(r^2-u^2)^(1/2)...

> Int(1/sqrt(u^2-r^2),u)=int(1/sqrt(u^2-r^2),u);

Int(1/(sqrt(u^2-r^2)),u) = ln(u+sqrt(u^2-r^2))

> Int(1/sqrt(u^2+r^2),u)=int(1/sqrt(u^2+r^2),u);

Int(1/(sqrt(u^2+r^2)),u) = ln(u+sqrt(u^2+r^2))

> Int(1/(u^2+r^2),u)=int(1/(u^2+r^2),u);

Int(1/(u^2+r^2),u) = 1/r*arctan(u/r)

Note that Maple may give a different form for the answer than most tables. But the forms are equivalent, as the answers can be shown to differ by a constant. In this exercise, we will use Maple to complete the square for a quadratic, and then compute the integral. In the package student there is a command completesquare which makes the process less tedious. Let us work a simple example.

> q:=3*x^2+4*x-5;

q := 3*x^2+4*x-5

> with(student):

> completesquare(q,x);


Now, let us suppose we want to calculate Int(1/sqrt(3*x^2+4*x-5),x) . Using the completed square form above, we should substitute u = sqrt(3)*(x+2/3) and r = sqrt(19/3) , so that du = sqrt(3)*dx , and the integral converts to Int(1/sqrt(3)/sqrt(u^2-r^2),u) . Let us compute this integral in Maple.

> int(1/sqrt(3)/sqrt(u^2-r^2),u);


Then let us substitute for u and r to obtain a solution to the original problem.

> subs(u = sqrt(3)*(x+2/3),r = sqrt(19/3),1/3*sqrt(3)*ln(u+sqrt(u^2-r^2)));


Let us see what Maple would have come up with if we just asked it to evaluate the original integral.

> int(1/sqrt(3*x^2+4*x-5),x);


A little thought reveals that these answers are the same.


For the integrals below do the following.

a) Complete the square for the quadratic expression in the integrand.

b) Write down the u and r substitutions necessary to convert the integral into one of the standard forms. Also find du in terms of dx .

c) Compute the integral of the substituted expression using Maple.

d) Substitute for u and r in the solution to express your answer in terms of the original information.

e) Integrate the original integral directly in Maple and compare your answers.

1) Int(1/sqrt(4*x^2-4*x+9),x)

2) Int(1/(x^2-4*x+7),x)

Submission worksheet: