#### To determine

**To evaluate: **

∫142+x2x dx

#### Answer

**Answer: **

∫142+x2x dx=825

#### Explanation

**1) Concept:**

i) 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.

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

**2) Calculation: **

∫142+x2x dx

After separating the denominator, the above integration becomes

∫14( 2x+ x2x )

By separating the integration,

∫142xdx+∫14x2xdx

Take the inverse of the denominator to simplify the equation

∫142·x-12dx+ ∫14x2-12 dx

By using the antiderivative of each term,

2x1/21/24 1+x5/25/241

Applying the Fundamental Theorem of Calculus,

241/21/2-11/21/2+[45/25/2-15/25/2]

By simplifying,

2221212-112+225252-152

After simplifying,

24-2+2525-1=4+2531=825

**Conclusion:**

Therefore,

∫142+x2x dx=825