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);

3*(x+2/3)^2-19/3

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);

1/3*sqrt(3)*ln(u+sqrt(u^2-r^2))

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)));

1/3*sqrt(3)*ln(sqrt(3)*(x+2/3)+sqrt(3*(x+2/3)^2-19/...

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);

1/3*ln(1/3*(3*x+2)*sqrt(3)+sqrt(3*x^2+4*x-5))*sqrt(...

A little thought reveals that these answers are the same.

Submission:

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:

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