   Chapter 7.1, Problem 48E

Chapter
Section
Textbook Problem

# (a) Prove the reduction formula ∫ cos n x   d x = 1 n cos n − 1 x sin x + n − 1 n ∫ cos n − 2 x   d x (b) Use part (a) to evaluate ∫ cos 2 x   d x .(c) Use parts (a) and (b) to evaluate ∫ cos 4 x   d x .

To determine

(a)

To prove: the reduction formula cosnxdx=1ncosn1xsinx+n1ncosn2xdx

Explanation

Proof:

Reduction formula is obtained by using the technique of integration by parts to solve the integral cosnxdx. The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation.

Formula used:

The formula for integration by parts in terms of u and v is given by

udv=uvvdu

Given:

The equation to prove, cosnxdx=1ncosn1xsinx+n1ncosn2xdx.

Calculation:

The integral can be rewritten as cosnxdx=cosn1xcosxdx

Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=cosn1x      dv=cosxdx

Then, the differentiation of u and antiderivative of dv will be

du=(n1)cosn2xsinxdx     v=sinx

Using the formula, the given integration will become

cosn1xcosxdx=cosn1xsinxsinx((n1)cosn2xsinx)dx=cosn1xsinx+(n1)sin2xcosn2xdx

Use the identity sin2x+cos2x=1 to substitute for sin2x:

cosn1xcosxdx=cosn1

To determine

(b)

To evaluate: the integral cos2xdx using reduction formula in part (a)

To determine

(c)

To evaluate: the integral cos4xdx using reduction formula and result from part (a)

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