Fundamental Theorem of Integral Calculus

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We use the fundamental theorem of integral calculus to find the derivative of a function defined as an integral.

1) Find the derivative of the functions f(x) defined by the following integral

f(x) = S [sin(t) dt]

df/dx =

NOTE: For S, there is an x on top and pi at the bottom.


2) Find the derivative of the functions f(x) defined by the following integral

f(x) = S [t^3 dt]

df/dx =

NOTE: For S, there is a 2x on top and x at the bottom.
 
for the first problem ...

\(\displaystyle \L \frac{d}{dx}[\int_a^x f(t) dt] = f(x)\), where a is a constant.


for the second problem ...

\(\displaystyle \L \frac{d}{dx}[\int_u^v f(t) dt] = f(v)\frac{dv}{dx} - f(u)\frac{du}{dx}\)

where u and v are functions of x.
 
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