you have: \begin{align}\int_{-\infty}^{+\infty}\frac{\cos^3x}{x^2+1}\mathrm dx&=\frac14\int_{-\infty}^{+\infty}\frac{\cos 3x+3\cos x}{1+x^2}\mathrm dx\\&=\frac18\int\left(\frac{e^{3ix}+e^{-3ix}+3e^{ix}+3^{-ix}}{1+x^2}\right)\mathrm dx\\&= \frac...
Even though integration allows us to calculate a large sum, sometimes, we really find it hard to derive the integration of a function. However, there is a solution to that. We can introduce a new and freestanding variable to find the integration. In the provided integral function ∫f(x), ...
Exponential functions, such as those in the form of e^x, have unique characteristics in calculus. Learn how to calculate the integrals of...
These get their name because they were first studied by mathematicians looking to calculate the arc length of an ellipse. The first recorded study of this problem was in 1655 by John Wallis and shortly after by Isaac Newton, who both published an infinite series expansion that gave the arc le...
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to calculate $\,\hat f(\omega)$, by assuming $\omega\le 0$ and with the real axis and the (infinite) semicircular arc in the upper half plane as contour of integration. How does the integrand behave on that arc, parametrised by $Re^{i\varphi}$ with $R\gg 0$ and $0\le\varphi...