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Handout Monday 11/24
Let’s see how useful the Fundamental Theorem of Calculus can be. The theorem tells us the relationships between the two main ideas in Calculus: derivatives and integrals. But one way to use the theorem is to evaluate definite integrals exactly by finding antiderivatives.
Remember what this looks like with an example. Introduce the notation “evaluated between”:
\begin{equation*}
\left.\int_2^6 x^3 \, dx = \frac{x^4}{4}\right|_2^6 = \frac{6^4}{4} - \frac{2^4}{4} = 324 - 4 = 320\text{.}
\end{equation*}
To use this, we need to find antiderivatives. Create a chart of standard antiderivatives. These are just our standard basic derivative rules written in reverse order. Basically.
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\(f(x) = x^n\) has antiderivative \(F(x) = \frac{x^{n+1}}{n+1}\text{.}\) This works for all \(n \ne -1\text{.}\)
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\(f(x) = e^x\) has antiderivative \(F(x) = e^x\text{.}\) What about other bases for exponential? \(f(x) = 3^x\) has antiderivative \(F(x) = \frac{3^x}{\ln(3)}\text{.}\)
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Trig functions: \(f(x) = \sin(x)\) has antiderivative \(F(x) = -\cos(x)\text{.}\) For \(f(x) = \cos(x)\text{,}\) an antiderivative is \(F(x) = \sin(x)\text{.}\) What about other trig functions? Note that we don’t want to worry about this, but we can think about what derivatives we get to find new anti-derivative rules. For example, what is the antiderivative of \(\frac{1}{\sqrt{1-x^2}}\text{.}\)
Now use these to evaluate integrals.
