Integrals 215 problem 05.mws

1. Checking written work with Maple.

Consider the integral Int((x+1/x)^2,x = 1 .. 2) . First you should compute this integral by hand. Then we can use Maple to check our answers by writing

> Int((x+1/x)^2,x=1..2);

Int((x+1/x)^2,x = 1 .. 2)

> int((x+1/x)^2,x=1..2);

29/6

We can also check indefinite integrals: First compute Int(sqrt(x)*(x^2-1/x),x) by hand, and then check with maple using

the following commands.

> Int(sqrt(x)*(x^2-1/x),x);

Int(sqrt(x)*(x^2-1/x),x)

> int(sqrt(x)*(x^2-1/x),x);

2/7*x^(7/2)-2*sqrt(x)

Submission:

Work the following problems by hand, and then check with Maple. Submit your Maple work in the submission worksheet.

(a) Int(y^9-2*y^5+3*y,y = 0 .. 1) .

(b) Int(cos(theta)+2*sin(theta),theta = 0 .. Pi/2)

(c) Int(3/x,x = -exp(2) .. -exp(1))

(d) Int(1/sqrt(1-x^2),x = 0 .. .5)

(e) Int(cos(x)-2*sin(x),x)

Submission worksheet:

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2. Checking your work with Maple; The Substitution Rule.

The Substitution Rule is one of several integration techniques that you will learn this semester - and it is a very important one. Integration techniques are learned best through practice, so work a lot of problems. Maple does not include the "+C" in its answers to indefinite integral problems. On occasion the expression you obtain for an indefinite integral will look quite different than Maple's, this can be ok. It is ok, if when you subtract the two results you get a CONSTANT. For instance, you might write sin(x)^2 as an antiderivative to 2*sin(x)*cos(x) and someone else might write -cos(x)^2 . They are both correct since their difference is 1 - a constant.

Submission:

Work the following problems by hand, and then check with Maple. Submit your Maple work in the submission worksheet. If your work by hand looks different than Maple's answer then reconcile the differences by subtracting

the two answers to see if the result is a constant.

(a) Int(x^3*(1-x^4)^5,x)

(b) Int(1/((1-3*t)^4),t)

(c) Int(arctan(x)/(1+x^2),x)

(d) Int(exp(x)/(exp(x)+1),x)

(e) Int(arcsin(x)/sqrt(1-x^2),x = 0 .. 1/2)

Submission worksheet:

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