# 15.5: Appendix E- Calculus Review


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This e-book uses the following integrals: \begin{align*} & \int_{x'}^{x''} \!\! \dx = x''-x' \cr & \int_{x'}^{x''}\frac{\dx}{x} = \ln \left| \frac{x''}{x'} \right| \cr & \int_{x'}^{x''} \!\! x^a \dx = \frac{1}{a+1} \left[ (x'')^{a+1} - (x')^{a+1} \right] \qquad \tx{($$a$$ is a constant other than $$-1$$)} \cr & \int_{x'}^{x''}\!\!\frac{\dx}{ax+b} = \frac{1}{a}\ln\left|\frac{ax''+b}{ax'+b}\right| \qquad \tx{($$a$$ is a constant)} \end{align*} Here are examples of the use of the expression for the third integral with $$a$$ set equal to $$1$$ and to $$-2$$: \begin{align*} & \int_{x'}^{x''} \!\! x \dx = \frac{1}{2}\left[(x'')^2-(x')^2\right] \cr & \int_{x'}^{x''} \! \frac{\dx}{x^2} = -\left( \frac{1}{x''} - \frac{1}{x'} \right) \end{align*}

## E.4 Line Integrals

A line integral is an integral with an implicit single integration variable that constraints the integration to a path.

The most frequently-seen line integral in this e-book, $$\int\!p\dif V$$, will serve as an example. The integral can be evaluated in three different ways:

1. The integrand $$p$$ can be expressed as a function of the integration variable $$V$$, so that there is only one variable. For example, if $$p$$ equals $$c/V$$ where $$c$$ is a constant, the line integral is given by $$\int\!p\dif V=c\int_{V_1}^{V_2}(1/V)\dif V = c\ln(V_2/V_1)$$.
2. If $$p$$ and $$V$$ can be written as functions of another variable, such as time, that coordinates their values so that they follow the desired path, this new variable becomes the integration variable.
3. The desired path can be drawn as a curve on a plot of $$p$$ versus $$V$$; then $$\int\!p\dif V$$ is equal in value to the area under the curve.

This page titled 15.5: Appendix E- Calculus Review is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Howard DeVoe via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.