The half-angle formulas are central!

This essay explores the theoretical importance of half-angle formulas, revealing their often-overlooked significance and numerous applications in mathematics. The essay introduces novel generalizations and applications, offering fresh insights into established mathematical concepts, ultimately inviting readers to ponder the hidden potential and ubiquity of these formulas in mathematical contexts.

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

Exponential Substitution: A New Method for Integration

1. Transformation for Integrals Involving $\sqrt{(x + b)^2 + a^2}$:

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

• Where:
  • $\theta = \sinh^{-1}\left( \dfrac{x + b}{a} \right)$
  • $a > 0$

2. Transformation for Integrals Involving $\tan\left( \dfrac{\beta}{2} \right)$ and $\tan\left( \dfrac{\gamma}{2} \right)$:

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

• Where:
  • $\alpha = \cos^{-1}\left( \dfrac{x + b}{a} \right)$
  • $\beta = \csc^{-1}\left( \dfrac{x + b}{a} \right)$
  • $\gamma = \sec^{-1}\left( \dfrac{x + b}{a} \right)$
• For the alternating signs $\pm$:
  • Use the upper sign when $\dfrac{x + b}{a} \geq 1$
  • Use the lower sign when $0 \leq \dfrac{x + b}{a} \leq 1$

3. Transformation for Integrals Involving $\sqrt{(x + b)^2 - a^2}$ and $\dfrac{\sqrt{x + b - a}}{\sqrt{x + b + a}}$:

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

• Where:
  • $\alpha = \cos^{-1}\left( \dfrac{x + b}{a} \right)$
  • For the alternating signs $\mp$ and $\pm$:
    • Use the upper sign when $\dfrac{x + b}{a} \geq 1$
    • Use the lower sign when $0 \leq \dfrac{x + b}{a} \leq 1$

For applications, see here.