Example 1: Fundamental Theorem of Calculus Pt. 1

Compute the derivative:

$\frac{d}{\mathit{dx}}\underset{0}{\overset{x}{\int }}\frac{t}{1+t^2}\mathit{dt}$

Step 1: Ensure that the lower bound of your antiderivative is a constant, a , and the upper bound is an equation, x .

Here the antiderivative is setup as it needs to be the lower bound is a constant, 0 , and the upper bound is an equation, x .

$\underset{0}{\overset{x}{\int }}\frac{t}{1+t^2}\mathit{dt}$

Step 2: Take the derivative of your antiderivative.

At this point you essentially cancel out the antiderivative and plug your upper bound equation , x , into the equation that was inside the antiderivative, $f\left(t\right)$.

Here you would replace all the t ’s in your equation with x ’s, the upper bound equation.

$\frac{d}{\mathit{dx}}\underset{0}{\overset{x}{\int }}\frac{t}{1+t^2}\mathit{dt}\mathit{=}\frac{x}{1+x^2}$

Step 3 (only if your equation is more than just an x ):
Here you can skip Step 3, the equation in this example is just an x .

Final Result:

The derivative of the equation $F\left(x\right)=\underset{0}{\overset{x}{\int }}\frac{t}{1+t^2}\mathit{dt}$ would be $F\prime \left(x\right)=\frac{x}{1+x^2}$.