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# Integration techniques summary and review

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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?