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Emmanuel José García's user avatar
Emmanuel José García's user avatar
Emmanuel José García's user avatar
Emmanuel José García
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The half-angle formulas are central!

Context: The theoretical importance of half-angle formulas

And as a picture is worth a thousand words...

The arrows indicate that the identity at the tail of the arrow implies the identity at the head of the arrow.

Integration via Exponential Substitution

We introduce Exponential Substitution, which is a transformation that simplifies certain composite trigonometric integrals and offers an alternative approach to integrating through trigonometric, hyperbolic, and Euler substitutions. For a more detailed description of how this technique works, visit the blog post 'Integration Using Some Euler-like Identities.' Examples 2, 3, and 7 illustrate how this method could be more efficient than the traditional approaches.

$$\boxed{\int f\left(x,\tan{\frac{\beta}{2}}, \tan{\frac{\gamma}{2}} \right)\,dx=\int f\left(\frac{e^{i\alpha}+e^{-i\alpha}}{2}a, e^{\pm\text{i}\alpha}, \frac{1-e^{\pm\text{i}\alpha}}{1+e^{\pm\text{i}\alpha}}\right)\,\frac{e^{\mp\text{i}\alpha}-e^{\pm\text{i}\alpha}}{2i}a\,d\alpha}\tag{1}$$

Where $\alpha=\cos^{-1}\left(\frac{x}{a}\right)$, $\beta=\csc^{-1}\left(\frac{x}{a}\right)$ and $\gamma=\sec^{-1}\left(\frac{x}{a}\right).$

Also, you can use

$$\boxed{\int f\left(x, \sqrt{x^2-a^2}, \frac{\sqrt{x-a}}{\sqrt{x+a}}\right)\,dx= \int f\left(\frac{e^{i\alpha}+e^{-i\alpha}}{2}a, \frac{e^{\mp\text{i}\alpha}-e^{\pm\text{i}\alpha}}{2}a, \frac{1-e^{\pm\text{i}\alpha}}{1+e^{\pm\text{i}\alpha}}\right)\,\frac{e^{\mp\text{i}\alpha}-e^{\pm\text{i}\alpha}}{2i}a\,d\alpha}\tag{2}$$

For the alternating signs $\mp$, use the upper sign when $\frac{x}{a} \geq 1$, and the lower sign when $0 \leq \frac{x}{a} \leq 1$.

$$\boxed{\int f\left(x, \sqrt{x^2+a^2}\right)\, dx = \int f\left(\frac{e^{\theta}-e^{-\theta}}{2}a, \frac{e^{\theta}+e^{-\theta}}{2}a\right) \frac{e^{\theta}+e^{-\theta}}{2}a\, d\theta}\tag{3}$$

Where $\theta=\sinh^{-1}(\frac{x}{a})$ and $a>0$.

Although closely related to integration using Euler's formula, it is not exactly the same.

Discovery is the privilege of the child: the child who has no fear of being once again wrong, of looking like an idiot, of not being serious, of not doing things like everyone else. - A. Grothendieck

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