#### To determine

**To evaluate: ∫02(45t3-34t2+25t)dt**

#### Answer

**Answer: **

**∫0245t3-34t2+25tdt=2**

#### 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: **

**∫02(45t3-34t2+25t)dt**

By separating the integration,

∫02(45t3-34t2+25t)dt=∫0245t3dt-∫0234t2dt+∫0225tdt

By using the antiderivative of each term,

=45t4420-34t3320+25t2220

Applying the Fundamental Theorem of Calculus,

=45244-0-34233-0+25222-0

Finally by simplifying we have

∫02(45t3-34t2+25t)dt=454-3483+252=165-2+45=4-2=2

**Conclusion:**

Therefore,

∫0245t3-34t2+25tdt=2