Question

F(x) = integrate t ^ 4 dt from 3 to x

Answer 1

We have the function f(t) defined as:

`f(x)=\int_3^xt^4dt`

We have to find f'(x).

We can solve this integral as:

`f(x)=(t^5)/(5)|^x_3=(x^5)/(5)-(3^5)/(5)`

If we derive this expression for f(x) we obtain:

`f^(\prime)(x)=(d)/(dx)((x^5)/(5)-(3^5)/(5))=(5)/(5)x^4+0=x^4`

NOTE: This expression could have been derived from the function inside the integral.

We can now find f'(3) as:

`f^(\prime)(3)=3^4=81`

Answer:

f'(x) = x^4

f'(3) = 81