Indefinite Integration

The following quote from Reference [1] can clarify a lot of confusion about indefinite integration: Today the process of finding the fluent of a given fluxion is called indefinite integration, or antidifferentiation, and…

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Multiple Cycles of the Complex Exponential Function

A Definite Integral of the Complex Exponential Function Recall from this post that: $ \int_0^{2 \pi} d\theta \cos\theta + i \int_0^{2 \pi} d\theta \sin\theta = 0$ and $ \int_0^{2 \pi} d\theta e^{i \theta}  = 0.$…

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How to Integrate in a Spherical Coordinate System

Review of Integration Integration with Cartesian coordinates is simple. The general form is $\int\int\int f(x,y,z)dxdydz$ in which $f(x,y,z)$ is an arbitrary function of the Cartesian coordinates. However, there may be cases in which integrating with…

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The Crux of Calculus

Define $\Delta x \equiv x_2 – x_1$, to be consistent with this post. Similarly, define $\Delta y \equiv y_2 – y_1$ and $\Delta z \equiv z_2 – z_1$. The Cartesian coordinates are $x$, $y$,…

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