Dated: 30-10-2024
Definite Integral
Wallis Sine Formula
When \(n\) is even
\[= \frac{n - 1}{n} \cdot \frac{n - 3}{n - 2} \cdot \ldots \cdot \frac 3 4 \cdot \frac 1 2 \cdot \frac \pi 2\]
When \(n\) is odd
\[\int_0^{\frac \pi 2} \sin^n(x)dx = \int_0^{\frac \pi 2} \cos^n(x)dx = \prod_{i = 0}^{\frac {n - 3} 2} \frac{n - 2i - 1}{n - 2i}\]
\[= \frac{n - 1}{n} \cdot \frac{n - 3}{n - 2} \cdot \ldots \cdot \frac 6 7 \cdot \frac 4 5 \cdot \frac 2 3\]
Integration by Parts
\[\int UVdx=U\int Vdx-\int\left(\int Vdx\cdot\frac{dU}{dx}\right)dx\]
Line Integrals
Let us say we have a curve \(C\) defined by \(f(x)\) and \(P\) be a point on this curve.
The position vector1 for \(P\) with respect to origin will be \(\vec P\) or let's say \(\vec r\).
Similarly, now imagine there is another point \(Q\) with position vector1 \(\vec Q\)
where \(\vec Q = \vec P + \vec {PQ}\).
Let us say that \(\vec {PQ} = \vec dr\)
If we divide the curve \(f(x)\) into multiple sections then the length of whole curve will be a sum of lengths of each section.
\[\sum_{p = 1}^n dr_p\]
Here \(p\) is just an index.
\[\lim_{dr \to 0} \sum_{p = 1}^n dr_p = \int_c dr\]
References
Read more about notations and symbols.