   Chapter 4.5, Problem 31E

Chapter
Section
Textbook Problem

# Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative. (take C = 0 ). ∫ x ( x 2 − 1 ) 3 d x

To determine

To evaluate:

The given indefinite integral xx2-13dx.

Explanation

1) Concept:

i) The substitution rule

If u=g(x) is a differentiable function whose range is I and f is continuous on I, then f(gx)g'xdx=f(u)du. ii) Indefinite integral

xn dx=xn+1n+1+C  n-1

iii)

abcfxdx=cabfxdx

2) Given:

xx2-13dx

3) Calculation:

The given integral is xx2-13dx

Here, use the substitution method.

Substitute x2-1=u.

Differentiating with respect to x.

2xdx=du

xdx=du2

Therefore, the given integral becomes

xx2-13dx=u3du2

=12u3du

Using concept ii),

=12u3+13+1+C

=12u44+C

=

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