FUNDAMENTAL THEOREM OF CALCULUS (PART-1)
If f(x) is a continuous function on [a,b], then the function g(x) defined by
g(x)=x∫af(t)dt,a≤x≤b
is an antiderivative of f, that is
g′(x)=f(x)orddx⎛⎝x∫af(t)dt⎞⎠=f(x).
Know about fundamental theorem of calculus Part 2 click here.
EXAMPLE:
Find the derivative of the function f(x)=x2∫0√1+t2dt
Solution:
Since the upper limit of integration is not x, we apply the chain rule. Let u=x2, then u′=2x.
Consider the new function
h(u)=u∫0√1+t2dt.
By the FTC1, we can write
h′(u)=√1+u2.
As f(x)=h(x2), we have
f′(x)=[h(x2)]′=h′(x2)⋅(x2)′=√1+(x2)2⋅2x=2x√1+x4.
HOMEWORK:
1. Find the derivative of the function
g(x)=x2∫3dtt.