#### To determine

**To evaluate:**

The given indefinite integral ∫xx2-13dx.

#### Answer

∫xx2-13dx= 18(x2-1)4+C

From the graph, it is reasonable to say 18(x2-1)4+C is the antiderivative of xx2-13.

#### 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=c∫abfxdx

**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

=12∫u3du

Using concept ii),

=12u3+13+1+C

=12u44+C

=18u4+C

Resubstitute x2-1=u,

Hence, the solution is

∫xx2-13dx=18(x2-1)4+C

Now, take C=0 and sketch the graph of the function fx=xx2-13 and its antiderivative Fx=18(x2-1)4.

The blue graph is for Fx=18(x2-1)4 and the red graph is for fx=xx2-13

The answer is reasonable since where f(x) is positive, there Fx is increasing; where f(x) is negative, there Fx is decreasing; where f(x) changes from negative to positive, there Fx has a local minimum; and where f(x) changes from positive to negative, there Fx has a local maximum.

**Conclusion:**

Therefore,

∫xx2-13dx= 18(x2-1)4+C

From the graph, it is reasonable to say 18(x2-1)4+C is the antiderivative of xx2-13