Integrals 215 problem 13.mws

1. Reduction formulas.

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

int(sin(x)^m*cos(x)^n,x) = cos(x)^(n-1)*sin(x)^(m+1...

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 dv = sin(x)^m*cos(x)*dx and u = cos(x)^(n-1) . Then we proceed as follows.

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

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

> v:=int(dv,x);

v := sin(x)^(m+1)/(m+1)

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

u := cos(x)^99

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

du := -99*cos(x)^98*sin(x)

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

Int(cos(x)^100*sin(x)^m,x) = cos(x)^99*sin(x)^(m+1)...

Submission worksheet:

Use the method above to establish the reduction formulas below

1) Int(x^n*cos(x),x) = x^n*sin(x)-n*Int(x^(n-1)*sin(x)...

2)

Int(x^n*exp(x),x) = x^n*exp(x)-n*Int(x^(n-1)*exp(x)...

Submission worksheet:

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2. Integrating products of sines and cosines.

At first, an integral like Int(sin(3*x)*cos(5*x),x) 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.

sin(m*x)*cos(n*x) = (sin((m-n)*x)+sin((m+n)*x))/2

sin(m*x)*sin(n*x) = (cos((m-n)*x)-cos((m+n)*x))/2

cos(m*x)*cos(n*x) = (cos((m-n)*x)+cos((m+n)*x))/2

Using the first formula above, we obtain that Int(sin(3*x)*cos(5*x),x) = Int((sin(-2*x)+sin(8*x))... . Let us compute both integrals using Maple, and see if they agree.

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

-1/16*cos(8*x)+1/4*cos(2*x)

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

-1/16*cos(8*x)+1/4*cos(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) Int(sin(2*x)*cos(4*x),x)

2) Int(sin(3*x)*sin(5*x),x)

3) Int(cos(4*x)*cos(6*x),x)

Submission worksheet:

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