L13-intTechReview.mws

__Calculus II__

**Lesson 13: **
**Integration Techniques Summary and Review**

You can use the student package in Maple to practice your integration techniques. First load the student package by typing

`> `
**with(student):**

Then read over the help screens on changevar, intparts, and value, paying particular attention to the examples at the bottom of the screens. Here are some examples.

** A Substitution Problem:**

Integration by substitution
is based on the chain rule. Thus if we have an integral which looks like
, then by make the change of variable
, and letting
we have a new, perhaps simpler integral
, to work on.

In Maple this is accomplished using the word
changevar
from the student package.

Find an antiderivative of

`> `
**F := Int(1/sqrt(1+sqrt(x)),x);**

Let's try the
change of variable
sqrt(x) = u .

`> `
**G := changevar(sqrt(x)=u,F);**

This does not seem to help. Lets try 1 + sqrt(x)= u

`> `
**G := changevar(1+sqrt(x)=u,F);**

Now we can do it by inspection, so just finish it off.

`> `
**G := value(G);**

Now substitute back and add in the constant.

`> `
**F := subs(u=sqrt(x),G) + C;**

Integration by substitution is the method use try after you decide you can't find the antiderivative by inspection.

**An Integration by Parts Problem: **

Integration by parts
is based on the product rule for derivatives. It is usually written
. It turns one integration problem into one which 'may' be more doable. Once you decide to use parts, the problem is what part of the integrand to let be u.

** **
Integrate

`> `
**F := Int(x^2*arctan(x),x);**

The word is
intparts
. Let's try letting
.

`> `
**G := intparts(F,x^2);**

That was a bad choice. Try letting

`> `
**G := intparts(F,arctan(x));**

This is much more promising. Split off the integral on the end.

`> `
**H := op(2,G);**

Now do a partial fractions decomposition of the integrand of H, using
parfrac
.

`> `
**H:= Int(convert(integrand(H),parfrac,x),x); **

Now we can do it by inspection.

`> `
**H1 := 1/6*x^2 - 1/3*1/2*ln(1+x^2);**

Let's check this with the student
value
.

`> `
**simplify(value(H-H1));**

Note the difference of a constant, which is fine for antiderivatives.

ETAIL
: The problem of choosing which part of the integrand to assign to u can often be solved quickly by following the etail convention. If your integrand has an Exponential factor, choose that for u, otherwise if it has a Trigonometric factor, let that be u, otherwise choose an Algebraic factor for u, otherwise chose an Inverse trig function, and as a last resort choose u to be a logarithmic factor. Let dv be what's left over.

**A Trig Substitution:**

Find an antiderivative of

`> `
**F := Int(x^3/sqrt(x^2+1),x);**

The presence of
suggests letting
.

`> `
**G := changevar(x=tan(t),F,t);**

Now use the trig identity
.

`> `
**G := subs(sqrt(1+tan(t)^2)=sec(t),G);**

Another substitution into the integrand.

`> `
**G := subs(tan(t)^3 = (sec(t)^2-1)*tan(t),G);**

Let's make a change of variable,

`> `
**H := changevar(sec(t)=u,G);**

From here, we can do it by inspection.

`> `
**H := value(H);**

Now unwind the substitutions.

`> `
**G := subs(u=sec(t),H);**

`> `
**F := subs(t = arctan(x),G);**

`> `
**F := subs(sec(arctan(x))=sqrt(1+x^2),F) + C;**

Checking this calculation:

`> `
**F1 := int(x^3/sqrt(x^2+1),x);**

It looks different, but is it?

`> `
**simplify(F-F1);**

Yes, but only by a constant.

**A Partial Fractions Problem**

Integrate the rational function

`> `
**y :=(4*x^2+x -1 )/(x^2*(x-1)*(x^2+1));**

First get the
partial fractions decomposition
of y.

`> `
**y := convert(y,parfrac,x);**

We can almost do this by inspection, except for the last term.

`> `
**F := Int(y,x); **

`> `
**F := expand(F);**

Now we can do each one by inspection. So we'll just use value .

`> `
**F := value(F) + C;**

**Exercises:**

Exercise: Use the student package to perform the following integrations.

Exercise: Find the area of the region enclosed by the x-axis and the curve
on the interval
. Sketch the region. Then find the vertical line
that divides the region in half and plot it.

Exercise: Find the length of the graph of the parabola
from O(0,0) to P(10,100). Find the point
on the graph which is 10 units from O along the graph. Make a sketch, showing the points O, P, and Q on the graph.

Exercise: Find the volume of the solid of revolution obtained by revolving the region trapped between the the graph of
on
and the x-axis about the x-axis. Sketch a graph. Does this volume approach a finite limit as n gets large?