In many books, there are tables of integrals which contain reduction formulas that allow you to reduce an integral to a simpler form. For example, the reduction formula

is very useful in integrating a product of sines and cosines when the power n of the cosine is an odd integer. Let us use integration by parts to establish this formula. Set and . Then we proceed as follows.

> dv:=sin(x)^m*cos(x);

> v:=int(dv,x);

> u:=cos(x)^(n-1);

> du:=simplify(diff(u,x));

> Int(simplify(u*dv),x)=u*v-Int(simplify(v*du),x);

Submission worksheet:

Use the method above to establish the reduction formulas below

1)

2)

Submission worksheet:

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At first, an integral like may appear complicated, but it is actually very simple to compute if you use the product-to-sum formulas from trigonometry. Let us recall the formulas that will make it simple to do integrals of products of sine and cosine functions.

Using the first formula above, we obtain that . Let us compute both integrals using Maple, and see if they agree.

> int(sin(3*x)*cos(5*x),x);

> int((sin(-2*x)+sin(8*x))/2,x);

Submission:

For the integrals below,

a) Use the product-to-sum formulas above to convert the integrand to a sum of sines and cosines.

b) Integrate the resulting simplified integrand in Maple.

c) Integrate the original function.

d) Compare the two results.

1)

2)

3)

Submission worksheet:

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>

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