91Ó°ÊÓ

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \int_{0}^{\infty} x^{3} e^{-x} d x $$

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \operatorname{erf}\left(\frac{1}{2}\right) $$

The following expression occurs in statistical mechanics: $$ P=\frac{n !}{(n p+u) !(n q-u) !} p^{n, \varphi+u} q^{n q-u}. $$ Use Stirling's formula to show that $$ \frac{1}{P} \sim x^{n p x} y^{n q y} \sqrt{2 \pi n p q x y}, $$ where \(x=1+\frac{u}{n p}, y=1-\frac{u}{n q}\), and \(p+q=1\). Hint: Show that $$ (n p)^{n D+u}(n q)^{n q-u}=n^{n} p^{n D+w} q^{n q-u} $$ and divide numerator and denominator of \(P\) by this expression.

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \int_{0}^{1} \sqrt{\frac{4-3 x^{2}}{1-x^{2}}} d x $$

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \int_{-\pi / 4}^{3 \pi / 4} \frac{d \phi}{\sqrt{1+\cos ^{2} \phi}} $$

Express the following integrals as \(\Gamma\) functions and evaluate them using a table of \(\Gamma\) functions. $$ \int_{0}^{\alpha} x^{2} e^{-x^{2}} d x \quad \text { Hint: Put } x^{2}=t $$

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \int_{0}^{\pi / 2} \frac{d x}{\sqrt{\cos x}} $$

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \int_{0}^{\pi / 2} \frac{d x}{\sqrt{2-\sin ^{2} x}} $$

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \frac{d}{d u}(\operatorname{cn} u) $$

Express the following integrals as \(\Gamma\) functions and evaluate them using a table of \(\Gamma\) functions. $$ \int_{0}^{1} \sqrt[3]{\ln x} d x $$