# The Fundamental Theorem of Calculus

## Key Questions

• If we can find the antiderivative function $F \left(x\right)$ of the integrand $f \left(x\right)$, then the definite integral ${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ can be determined by $F \left(b\right) - F \left(a\right)$ provided that $f \left(x\right)$ is continuous.

We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that $f \left(x\right)$ is continuous and why.

FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work.

For most students, the proof does give any intuition of why this works or is true. But let's look at $s \left(t\right) = {\int}_{a}^{b} v \left(t\right) \mathrm{dt}$. We know that integrating the velocity function gives us a position function. So taking $s \left(b\right) - s \left(a\right)$ results in a displacement.

• Fundamental Theorem of Calculus

$\frac{d}{\mathrm{dx}} {\int}_{a}^{x} f \left(t\right) \mathrm{dt} = f \left(x\right)$

This theorem illustrates that differentiation can undo what has been done to $f$ by integration.

Let us now look at the posted question.

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \setminus {\int}_{1}^{x} \sqrt{{e}^{t} + \sin t} \mathrm{dt} = \sqrt{{e}^{x} + \sin x}$

I hope that this was helpful.

• ${\int}_{a}^{b} f \left(x\right) \mathrm{dx} = F \left(b\right) - F \left(a\right)$,
where F is an antiderivative of $f$