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

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:**

`> `

`> `

`> `