Find the constants ## A ## and ## B ##

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In summary, the constants A and B are used to represent coefficients in a mathematical equation and can help us understand the relationship between variables and make predictions. They are found by using a mathematical equation and a set of data points, and can be affected by factors such as equation type, data quality, and assumptions. The values of A and B can change over time and can provide insights into the data, make predictions, and be used in further analysis.
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Math100
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Homework Statement
Define the logarithmic integral ## li(x) ## by ## li(x)=\int_{2}^{x}\frac{dt}{\log t} ##, for ## x>2 ##. Prove that ## li(x)=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}+A ## and ## li(x)=\frac{x}{\log x}+\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}+B ##, for some constants ## A ## and ## B ## that you should determine.
Relevant Equations
None.
Proof:

Observe that
\begin{align*}
&li(x)=\int_{2}^{x}\frac{dt}{\log t}\\
&=[\frac{t}{\log t}]_{2}^{x}-\int_{2}^{x}t(\frac{1}{\log t})'dt\\
&=\frac{x}{\log x}-\frac{2}{\log 2}-\int_{2}^{x}t(-\frac{1}{t\log^2 t})dt\\
&=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}-\frac{2}{\log 2}\\
&=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}+A\\
\end{align*}
where ## A=-\frac{2}{\log 2} ##.
This implies ## li(x)=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}-\frac{2}{\log 2} ##.
Since
\begin{align*}
&\int_{2}^{x}\frac{dt}{\log^2 t}=[\frac{t}{\log^2 t}]_{2}^{x}+2\int_{2}^{x}\frac{dt}{\log^3 t}\\
&=\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}-\frac{2}{\log^2 2},\\
\end{align*}
it follows that ## li(x)=\frac{x}{\log x}+\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}-\frac{2}{\log 2}-\frac{2}{\log^2 2} ##.
Thus ## li(x)=\frac{x}{\log x}+\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}-(\frac{2}{\log 2}+\frac{2}{\log^2 2}) ##, where ## B=-\frac{2}{\log 2}-\frac{2}{\log^2 2} ##.
Therefore, ## li(x)=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}+A ## and
## li(x)=\frac{x}{\log x}+\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}+B ##
where ## A=-\frac{2}{\log 2} ## and ## B=-\frac{2}{\log 2}-\frac{2}{\log^2 2} ##.
 
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  • #2
Math100 said:
Homework Statement:: Define the logarithmic integral ## li(x) ## by ## li(x)=\int_{2}^{x}\frac{dt}{\log t} ##, for ## x>2 ##. Prove that ## li(x)=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}+A ## and ## li(x)=\frac{x}{\log x}+\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}+B ##, for some constants ## A ## and ## B ## that you should determine.
Relevant Equations:: None.

Proof:

Observe that
\begin{align*}
&li(x)=\int_{2}^{x}\frac{dt}{\log t}\\
&=[\frac{t}{\log t}]_{2}^{x}-\int_{2}^{x}t(\frac{1}{\log t})'dt\\
&=\frac{x}{\log x}-\frac{2}{\log 2}-\int_{2}^{x}t(-\frac{1}{t\log^2 t})dt\\
&=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}-\frac{2}{\log 2}\\
&=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}+A\\
\end{align*}
where ## A=-\frac{2}{\log 2} ##.
This implies ## li(x)=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}-\frac{2}{\log 2} ##.
Since
\begin{align*}
&\int_{2}^{x}\frac{dt}{\log^2 t}=[\frac{t}{\log^2 t}]_{2}^{x}+2\int_{2}^{x}\frac{dt}{\log^3 t}\\
&=\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}-\frac{2}{\log^2 2},\\
\end{align*}
it follows that ## li(x)=\frac{x}{\log x}+\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}-\frac{2}{\log 2}-\frac{2}{\log^2 2} ##.
Thus ## li(x)=\frac{x}{\log x}+\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}-(\frac{2}{\log 2}+\frac{2}{\log^2 2}) ##, where ## B=-\frac{2}{\log 2}-\frac{2}{\log^2 2} ##.
Therefore, ## li(x)=\frac{x}{\log x}+\int_{2}^{x}\frac{dt}{\log^2 t}+A ## and
## li(x)=\frac{x}{\log x}+\frac{x}{\log^2 x}+2\int_{2}^{x}\frac{dt}{\log^3 t}+B ##
where ## A=-\frac{2}{\log 2} ## and ## B=-\frac{2}{\log 2}-\frac{2}{\log^2 2} ##.
Looks good. The formula for integration by parts is
$$
\int_a^b u'(x)v(x)\,dx=\left[u(x)v(x)\right]_a^b - \int_a^b u(x)v'(x)\,dx
$$
and you applied it to ##u'(x)=1## twice. If you quote the formula under "relevant equations" and then simply note ##u'=1## then it is a bit easier to read.

A way to remember the formula is to note that it is at its core the Leibniz rule of differentiation:
\begin{align*}
(u\cdot v )'&=u'v +uv'\\
u'v&=(u\cdot v )' - uv'\\
\int u'v &=\int [(u\cdot v )' -uv']\\
\int u'v &=uv-\int uv'
\end{align*}

The quotient rule of differentiation is also simply the Leibniz rule:
$$
\left(\dfrac{u}{v}\right)'=(u\cdot v^{-1})'=u'v^{-1}+u\left(v^{-1}\right)'=\dfrac{u'v}{v^2}-\dfrac{uv'}{v^2}=\dfrac{u'v-uv'}{v^2}
$$

In other parts of mathematics, the Leibniz rule takes the form of the Jacobi-identity, or the defining equation for derivations. All these formulas are simply the Leibniz rule ##(u\cdot v )'=u'v +uv'## which is all one has to remember.
 
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FAQ: Find the constants ## A ## and ## B ##

What is the purpose of finding the constants A and B?

The constants A and B are used to represent the numerical values of unknown quantities in a mathematical equation or formula. Finding these constants allows for the accurate and precise prediction of future outcomes or behavior of a system.

How do you determine the values of A and B?

The values of A and B can be determined through a variety of methods, such as experimentation, data analysis, or mathematical calculations. The specific method used will depend on the context and nature of the problem at hand.

Can A and B be negative or complex numbers?

Yes, A and B can be negative or complex numbers. In fact, in many cases, these constants may need to be negative or complex in order to accurately represent the behavior of a system or phenomenon.

Is there a specific formula or algorithm for finding A and B?

There is no one-size-fits-all formula or algorithm for finding A and B. The approach to finding these constants will vary depending on the specific problem and the available data or information.

What are some common applications of finding the constants A and B?

Finding the constants A and B is a crucial step in many scientific and mathematical fields, such as physics, chemistry, engineering, and statistics. Some common applications include predicting the behavior of physical systems, analyzing data and trends, and developing mathematical models for various phenomena.

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