#### To determine

**To find:**

Function f and number a

**Solution: f=xx, a=9**

#### Explanation

**1) Concept:**

i) ddx∫axftdt=f(x)

**2) Calculations:**

Consider,

6+∫axftt2dt=2x

Differentiating on both sides with respect to x

ddx6+ddx∫axftt2dt=ddx2x

By using concept (i),

0+fxx2=2·12x

Multiplying x2 to both sides

fx=x2x

By simplifying, theabove equation becomes

fx=xx

Now, to find a,

Substitute ft=tt in the given equation

6+∫axttt2dt=2x

On simplifying above equation, we get,

6+∫ax1tdt=2x

The anti derivative of 1t is 2t. By using fundamental theorem of calculus,

6+2txa=2x

Thus

6+2x-2a=2x

Subtracting 2x from both sides

6=2a

Dividing 2 to both sides

3=a

By taking the square roots on both sides

9=a

**Conclusion:**

Therefore, f(x)=xx, a=9