22. Riemann Sums, Integrals and the FTC

d. Integration Rules and Properties

2. Exp and Log Functions

The table on the previous page listed the properties of derivatives and integrals with respect to algebraic operations. The tables on this page give the derivatives and integrals for exponential and logarithmic functions.

Exponential and Logarithmic Functions
	Derivative Rule	Integral Rule
Natural Exponential	\(\dfrac{d}{dx}(e^x) =e^x\)	\(\displaystyle \int e^x\,dx=e^x+C\)
General Exponential	\(\dfrac{d}{dx}(b^x) =b^x\ln b\)	\(\displaystyle \int b^x\,dx=\dfrac{b^x}{\ln b}+C\)
Natural Logarithm\(^\text{1}\)	\(\dfrac{d}{dx}(\ln x) =\dfrac{1}{x}\)	\(\displaystyle \int \dfrac{1}{x}\,dx=\ln\|x\|+C\)
General Logarithm\(^\text{2}\)	\(\dfrac{d}{dx}(\log_b x) =\dfrac{1}{x\ln b}\)	\(\displaystyle \int \dfrac{1}{x}\,dx=\ln b\log_b\|x\|+C\)

\(^\text{1}\) This is the integral which produces \(\ln x\). We will discuss the integrals of \(\ln x\) and \(\log_b x\) in Calculus 2 in the chapter on Integration by Parts.
\(^\text{2}\) This integral formula is not useful, since the formula using \(\ln x\) is equivalent and easier.

Why is there an absolute value in the log integrals?

Like the function \(\sqrt{x}\), the function \(\ln{x}\) is only defined for positive values of \(x\), since it is the inverse function of \(e^x\) which is always positive. However, the function \(\dfrac{1}{x}\) is defined for all \(x\ne0\) and we might want to know \(\displaystyle \int \dfrac{1}{x}\,dx\) for negative as well as positive values of \(x\).

For \(x \gt 0\) we have \(\displaystyle \int \dfrac{1}{x}\,dx=\ln(x)+C\) without or with the absolute value.

However, for \(x \lt 0\), we need to recompute \(\displaystyle \int \dfrac{1}{x}\,dx\). We work backwards: For \(x \lt 0\), we have \(|x|=-x\) and so \(\ln|x|=\ln(-x)\). We differentiate using the Chain Rule: \[\begin{aligned} \dfrac{d}{dx}\ln|x|&=\dfrac{d}{dx}\ln(-x)=\dfrac{1}{(-x)}\dfrac{d}{dx}(-x) \\ &=\dfrac{1}{(-x)}(-1)=\dfrac{1}{x} \end{aligned}\] Reversing this, we have \(\displaystyle \int \dfrac{1}{x}\,dx=\ln|x|+C\). So this formula works for both positive or negative values of \(x\).

The following exercise is the same as on a previous page on antiderivative rules for exponential functions.

Compute \(\displaystyle \int 3x^2e^{\small x^{\scriptstyle 3}}\,dx\) and \(\displaystyle \int_0^1 3x^2e^{\small x^{\scriptstyle 3}}\,dx\).

\(\displaystyle \int 3x^2e^{\small x^{\scriptstyle 3}}\,dx =e^{\small x^{\scriptstyle 3}}+C\)
\(\displaystyle \int_0^1 3x^2e^{\small x^{\scriptstyle 3}}\,dx=e-1\)

Since \(\dfrac{d}{dx}x^3=3x^2\), the Chain Rule and Exponential Rule say the indefinite integral is: \[ \int 3x^2e^{\small x^{\scriptstyle 3}}\,dx=e^{\small x^{\scriptstyle 3}}+C \] Be sure to check by differentiating!

By the Fundamental Theorem of Calculus, the definite integral is: \[\begin{aligned} \int_0^1 &3x^2e^{\small x^{\scriptstyle 3}}\,dx =\left[e^{\small x^{\scriptstyle 3}}\right]_0^1 \\ &=\left(e^{\small 1^{\scriptstyle 3}}\right)-\left(e^{\small 0^{\scriptstyle 3}}\right) =e-1 \end{aligned}\]

We check the indefinite integral by differentiating. If \(F=e^{\small x^{\scriptstyle 3}}\) then: \[ F'(x)=e^{\small x^{\scriptstyle 3}}\cdot3x^2 \]

Compute \(\displaystyle \int \dfrac{8}{x}\,dx\) and \(\displaystyle \int_{-4}^{-2} \dfrac{8}{x}\,dx\).

\(\displaystyle \int \dfrac{8}{x}\,dx =8\ln|x|+C\)
\(\displaystyle \int_{-4}^{-2} \dfrac{\rule{0pt}{12pt}8}{x}\,dx=-8\ln2\)

\(8\) is a constant. So the integral is easy. Don't forget the absolute value! \[ \int \dfrac{8}{x}\,dx=8\ln|x|+C \]

By the Fundamental Theorem of Calculus, the definite integral is: \[\begin{aligned} \int_{-4}^{-2} \dfrac{8}{x}\,dx &=\left[\rule{0pt}{10pt}8\ln|x|\right]_{-4}^{-2} =8\ln|-2|-8\ln|-4| \\ &=8\ln2-8\ln4 =8\ln2-16\ln2 =-8\ln2 \end{aligned}\] The answer is negative because \(\dfrac{8}{x}\) is negative on \([-4,-2]\).

22. Riemann Sums, Integrals and the FTC

d. Integration Rules and Properties

2. Exp and Log Functions

Why is there an absolute value in the log integrals?

Heading