#### To determine

**To evaluate:**

∫011+r3dr

#### Answer

**∫011+r3dr=154**

#### Explanation

**1) Concept:**

i) The Fundamental Theorem of Calculus (part 2): Suppose f is continuous on [a, b], then

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

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

**2) Formula**

a+b3=a3+3a2b+3ab2+b3

**3) Calculation: **

**∫011+r3dr**

Rewrite the integration by expanding the cubic term

∫011+3r+3r2+r3dr

By separating the integration

∫011dr+∫013r dr+∫013r2dr+∫01r3 dr

By using the antiderivative of each term,

r 10+3r2210+3r3310+[r44]10

By substituting limits

1-0+3122-0+113-0+(144-0)

By calculating

=1+32+1+14=2+74=154

**Conclusion:**

Therefore,

**∫011+r3dr=154**