   Chapter 8.1, Problem 97E

Chapter
Section
Textbook Problem

# Finding a Pattern(a) Find ∫ cos 3 x   d x .(b) Find ∫ cos 5 x   d x .(c) Find ∫ cos 7 x   d x .(d) Explain how to find ∫ cos 15 x   d x without actually integrating.

(a)

To determine

To calculate: The value of the integral given as, cos3xdx.

Explanation

Given:

The provided integral is cos3xdx.

Formula Used:

cos2x+sin2x=1

tndt=tn+1n+1+C

Calculation:

Consider,

cos3xdx

Split cos3x as shown,

cos3xdx=cos2xcosxdx …… (1)

By making use of cos2x+sin2x=1, equation (1) can be rewritten as shown,

cos3xdx=(1sin2x)(cosx)dx …… (2)

Integrate equation (2) by the method of substitution and let,

sinx=t …… (3)

Differentiate sinx=t on both the sides with respect to t,

(cosx)dx=dt …… (4)

Put (3) and (4) in equation (2),

(b)

To determine

To calculate: The value of the integral given as, cos5xdx will be.

(c)

To determine

To calculate: What the value of the integral given as, cos7xdx will be.

(d)

To determine

To calculate: What the solution of the integral given as, cos15xdx will be, without integrating it.

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