#### To determine

**To evaluate: ∫01(1-8v3+16v7)dv**

#### Answer

**∫01(1-8v3+16v7)dv=1**

#### Explanation

**1) Concept:**

i) The Fundamental Theorem of Calculus (part 2)

ii) Separate out the integration and then use antiderivative of each term

**2) Theorem:**

The Fundamental Theorem of Calculus: Suppose f is continuous on [a, b], then

∫abfxdx=Fb-F(a), where F is antiderivative of f, that is F'=f.

**3) Calculation: **

∫01(1-8v3+16v7)dv By separating the integral

∫01(1-8v3+16v7)dv=∫011dv-∫018v3dv+∫0116v7dv

By using the antiderivative of each term,

=v 10-8v4410+16v8810

Applying the Fundamental Theorem of Calculus

=1-0-21-0+2(1-0)

On simplifying,

=1-2+2=1

**Conclusion:**

Therefore,∫01(1-8v3+16v7)dv=1