Using the substitution (u = x^{\phi}), (du = \phi x^{\phi-1} dx), we get:
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: Treatment of definite integrals as the limit of a sum and their various properties. Geometric Applications golden integral calculus pdf
Determining the center of gravity and moments of inertia using integration. Advanced "Golden Calculus" Framework
You might wonder: why collect such integrals into a PDF? Using the substitution (u = x^{\phi}), (du =
Introduction to double and triple integrals.
[ \int_{0}^{\infty} \frac{dx}{\phi^{,x} \cdot \Gamma(x+1)} = 1 ] Geometric Applications Determining the center of gravity and
[ \int_{0}^{\infty} \frac{x^{a-1}}{1+x^{b}} , dx = \frac{\pi}{b \sin(\pi a / b)}, \quad 0<a<b ]
She clicked it. The first page was blank except for a single, hand-drawn-looking equation in the center: