If #F(x)=int_1^(x^2)sqrt(t^2+3)dt#, then #F'(2)=#?
I anti-differentiated this function and plugged the value in, I got #sqrt(19)^3*8/3-16/3# . However, the correct answer is #4sqrt(19)# .
Is my answer wrong or I need to simplify it? If so, how?
Thank you!
I anti-differentiated this function and plugged the value in, I got
Is my answer wrong or I need to simplify it? If so, how?
Thank you!
1 Answer
We need to use the Chain Rule in conjunction with FToC part I. See below.
Explanation:
This question requires application of the first part of the Fundamental Theorem of Calculus, which tells us if
This makes sense, it connects our knowledge of differentiation and integration together, telling us that the derivative of an integrated function is really just that function itself.
Here, however, our definite integral involves
Let
We see
where
Thus,
So,
Then,