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Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.

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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.

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Evaluate \[\int_{0}^{\sqrt{3}} (x+4)^2 e^{x^2} dx+\int_{\sqrt{3}}^{0} (x-4)^2 e^{x^2} dx.\]

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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 \(\lfloor -3.4 \rfloor = -4\).

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Determine the value of \( 100 \int_0^{\pi} \sqrt{ 1 + \sin x} \, dx \).

This problem is proposed by Cody.

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