Integration Techniques
When \(a=2\) and \(b=4,\) \( \int_{a}^{b}{\sqrt{(x-a)(b-x)}}dx \) is equal to \(\underline{\qquad}.\)
For \(n\) a positive integer, what is the indefinite integral \[\int{ x{(1-x)}^{n+6} }dx?\]
Details and assumptions
Use \(C\) as the constant of integration.
Evaluate \[\int_{0}^{\sqrt{3}} (x+4)^2 e^{x^2} dx+\int_{\sqrt{3}}^{0} (x-4)^2 e^{x^2} dx.\]
Given that \(\displaystyle \int_0^4 x^3\sqrt{9+x^2} dx = a\), what is the value of \(\lfloor a \rfloor\)?
Details and assumptions
Greatest Integer Function: \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer less than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -3.4 \rfloor = -4\).
Determine the value of \( 100 \int_0^{\pi} \sqrt{ 1 + \sin x} \, dx \).
This problem is proposed by Cody.