Identifier: Fundamental Theorem of Calculus Pt. 1

FTC1 is usually straightforward to identify as it can only be applied when its unique conditions are met.

If you are given an integral with the upper bound, the lower bound, or both the upper bound and the lower bound as an equation , then you are most likely going to apply the FTC1.  If you stop and think about it, the only time you ever see equations in the bounds of an integral is when you need to apply the FTC1.

• $\underset{a}{\overset{x}{\int }}f\left(t\right)\mathit{dt}$
• $\underset{x}{\overset{a}{\int }}f\left(t\right)\mathit{dt}$
• $\underset{–\mathit{sin}\left(x\right)}{\overset{\mathit{sin}\left(x\right)}{\int }}f\left(t\right)\mathit{dt}$

If you are given a function that is equal to an integral with equations in the bounds, $F\left(x\right)=\underset{a}{\overset{x}{\int }}f\left(t\right)\mathit{dt}$, and then being asked about the derivative, $F\prime \left(x\right)$, or being asked a question that only a derivative could answer.

• When is $F\left(x\right)$increasing or decreasing?
• Does $F\left(x\right)$ have a local max or local min?

If you are being asked to take the derivative of an antiderivative , then you are most likely going to apply the FTC1.

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