   Chapter 4.R, Problem 21E

Chapter
Section
Textbook Problem

# Evaluate the integral, if it exists. ∫ 0 1 v 2 cos ( v 3 )   d v

To determine

To evaluate:

01v2cos(v3)dv if it exists

Explanation

1) Concept:

By using substitution rule for definite integrals and fundamental rule of calculus

2) Theorem:

Fundamental theorem of calculus:

If f is continuous on [a, b], then abfxdx=Fb-F(a).

Substitution rule for definite integrals:

If g' is continuous on [a, b] and f is continuous on the range of u=g(x) then

abfgxg'xdx=g(a)g(b)f(u)du

3) Formula:

cosxdx=sinx+C

3) Given:

01v2cos(v3)dv

4) Calculation:

Consider, 01v2cos(v3)dv

Since the function is continuous on the given interval, the integral exists.

By applying substitution rule for definite integrals,

Let u=v3

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