If $J_r$ is the integral

$$\int_0^{\infty} x^r \exp \left(-x^2\right) d x$$

show that (a) $J_{2 r+1}=(r !) / 2$, (b) $J_{2 r}=2^{-r}(2 r-1)(2 r-3) \cdots(5)(3)(1) J_0$.

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 $\left(-\frac{3}{2}\right) !$

Show, using the following methods, that the indefinite integral of $x^3 /(x+1)^{1 / 2}$ is

$$J=\frac{2}{35}\left(5 x^3-6 x^2+8 x-16\right)(x+1)^{1 / 2}+c .$$

(a) Repeated integration by parts. (b) Setting $x+1=u^2$ and determining $d J / d u$ as $(d J / d x)(d x / d u)$.

Find the first derivatives of (a) $x /(a+x)^2$, (b) $x /(1-x)^{1 / 2}$, (c) $\tan x$, as $\sin x / \cos x$, (d) $\left(3 x^2+2 x+1\right) /\left(8 x^2-4 x+2\right)$.

Use inttegration by parts to evaluate the following: (a) $\int_0^y x^2 \sin x d x$; (b) $\int_1^y x \ln x d x$; (c) $\int_0^y \sin ^{-1} x d x$ (d) $\int_1^y \ln \left(a^2+x^2\right) / x^2 d x$.

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 $0<x \leq 1$.