Fundamental Theorem of Calculus Problem?

Question

Find the derivatives of the function

Integral from tan x to x^2 of dt/sqrt(4+t^4).

I understand this is a fundamental theorem of calculus problem, and I read another question that had a similar question posed, but the answer was too difficult for me to understand.

Thanks in advance.

Answer 1

The trick to these problems is that there is NO differentiation or integration involved; just the chain rule.

So if f(x) = Integral (tan(x) to x^2 of 1/sqrt(4+t^4))dt, when taking the derivative, all you have to do is plug in the upper integral bound for t, apply the chain rule, subtract the lower integral bound plugging in that value for t, and then apply the chain rule.

I'll show you what I mean:

f'(x) =[ 1/sqrt(4 + (x^2)^4) ] * 2x - [ 1/sqrt(4 + ((tanx)^2)^4) ] * (sec(x))^2

Note that the derivative of x^2 is 2x, and the derivative of tanx is (sec(x))^2 or "secant squared x".

Rule of thumb: When taking the derivative of an integral, just substitute your t with the function in the upper bound, apply the chain rule, subtract the function in the lower bound, apply the chain rule.

I repeat, NO integration and NO differentiation done. Just the chain rule.

Answer 2

ya puggy did it exactly right.

Also if you had numbers for the limits of integration you would've gotten zero (because the integral would've been a number).

And the other case is when there is a number for one of the limits of integration and a f(x) for another... then obviously there is only one term because the derivative of the number in the chain rule would be 0 times something.

Answer 3

Use Office Excel Program to solve this problem or visit these sites :

http://integrals.wolfram.com/index.jsp

http://mathworld.wolfram.com/