The gamma function $\Gamma(n)$ is defined for all $n>-1$ by

$$\Gamma(n+1)=\int_0^{\infty} x^n e^{-x} d x .$$

Find a recurrence relation connecting $\Gamma(n+1)$ and $\Gamma(n)$. (a) Deduce (i) the value of $\Gamma(n+1)$ when $n$ is a non-negative integer, and (ii) the value of $\Gamma\left(\frac{7}{2}\right)$, given that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$. (b) Now, taking factorial $m$ for any $m$ to be defined by $m !=\Gamma(m+1)$, evaluate $\left(-\frac{3}{2}\right) !$

Determine whether the following integrals exist and, where they do, evaluate them: (a) $\int_0^{\infty} \exp (-\lambda x) d x$; (b) $\int_{-\infty}^{\infty} \frac{x}{\left(x^2+a^2\right)^2} d x$; (c) $\int_1^{\infty} \frac{1}{x+1} d x$; (d) $\int_0^1 \frac{1}{x^2} d x$; (e) $\int_0^{\pi / 2} \cot \theta d \theta$; (f) $\int_0^1 \frac{x}{\left(1-x^2\right)^{1 / 2}} d x$.

By making the substitution $x=a \cos ^2 \theta+b \sin ^2 \theta$, evaluate the definite integrals $J$ between limits $a$ and $b(>a)$ of the following functions: (a) $[(x-a)(b-x)]^{-1 / 2}$; (b) $[(x-a)(b-x)]^{1 / 2}$ (c) $[(x-a) /(b-x)]^{1 / 2}$.

Find the indefinite integrals, $J$, of the following functions involving sinusoids: (a) $\cos ^5 x-\cos ^3 x$ (b) $(1-\cos x) /(1+\cos x)$; (c) $\cos x \sin x /(1+\cos x)$; (d) $\sec ^2 x /\left(1-\tan ^2 x\right)$.

Find the derivative of $f(x)=(1+\sin x) / \cos x$ and hence determine the indefinite integral $J$ of $\sec x$.

Use logarithmic integration to find the indefinite integrals $J$ of the following: (a) $\sin 2 x /\left(1+4 \sin ^2 x\right)$; (b) $e^x /\left(e^x-e^{-x}\right)$; (c) $(1+x \ln x) /(x \ln x)$; (d) $\left[x\left(x^n+a^n\right)\right]^{-1}$.

Find the integral $J$ of $\left(a x^2+b x+c\right)^{-1}$, with $a \neq 0$, distinguishing between the cases (i) $b^2>4 a c$, (ii) $b^2<4 a c$ and (iii) $b^2=4 a c$.

Express $x^2(a x+b)^{-1}$ as the sum of powers of $x$ and another integrable term, and hence evaluate

$$\int_0^{b / a} \frac{x^2}{a x+b} d x .$$

Find the indefinite integrals $J$ of the following ratios of polynomials: (a) $(x+3) /\left(x^2+x-2\right)$; (b) $\left(x^3+5 x^2+8 x+12\right) /\left(2 x^2+10 x+12\right)$ (c) $\left(3 x^2+20 x+28\right) /\left(x^2+6 x+9\right)$; (d) $x^3 /\left(a^8+x^8\right)$.

Find the following indefinite integrals: (a) $\int\left(4+x^2\right)^{-1} d x$; (b) $\int\left(8+2 x-x^2\right)^{-1 / 2} d x$ for $2 \leq x \leq 4$; (c) $\int(1+\sin \theta)^{-1} d \theta$; (d) $\int(x \sqrt{1-x})^{-1} d x$ for $0<x \leq 1$.

Find the first derivatives of (a) $x^2 \exp x$, (b) $2 \sin x \cos x$, (c) $\sin 2 x$, (d) $x \sin a x$, (e) $(\exp a x)(\sin a x) \tan ^{-1} a x$, (f) $\ln \left(x^a+x^{-a}\right)$, (g) $\ln \left(a^x+a^{-x}\right)$, (h) $x^x$.

Show that the curve $x^3+y^3-12 x-8 y-16=0$ touches the $x$-axis.

Use Rolle's theorem to deduce that, if the equation $f(x)=0$ has a repeated root $x_1$, then $x_1$ is also a root of the equation $f^{\prime}(x)=0$. (a) Apply this result to the 'standard' quadratic equation $a x^2+b x+c=0$, to show that a necessary condition for equal roots is $b^2=4 a c$. (b) Find all the roots of $f(x)=x^3+4 x^2-3 x-18=0$, given that one of them is a repeated root. (c) The equation $f(x)=x^4+4 x^3+7 x^2+6 x+2=0$ has a repeated integer root. How many real roots does it have altogether?

For the function $y(x)=x^2 \exp (-x)$ obtain a simple relationship between $y$ and $d y / d x$ and then, by applying Leibnitz' theorem, prove that

$$x y^{(n+1)}+(n+x-2) y^{(n)}+n y^{(n-1)}=0$$