Integrals 215 problem 06.mws

Integrals involving quadratic expressions of the form are often evaluated with the help of completing the square to convert it to a form that looks like either , or . 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(u^2-r^2),u)=int(1/sqrt(u^2-r^2),u);

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

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

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;

> with(student):

> completesquare(q,x);

Now, let us suppose we want to calculate . Using the completed square form above, we should substitute and , so that , and the integral converts to . 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.

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 in terms of .

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)

2)

Submission worksheet:

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