Improper integral to infinity with inverse trig

6 In this question you must show detailed reasoning.
Determine the exact value of \(\int _ { 9 } ^ { \infty } \frac { 18 } { x ^ { 2 } \sqrt { x } } \mathrm {~d} x\).

6 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).

2 In this question you must show detailed reasoning. Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 2 } - 4 x + 5 } d x\)

1. (a) Explain why

$$\int _ { \frac { 4 } { 3 } } ^ { \infty } \frac { 1 } { 9 x ^ { 2 } + 16 } d x$$ is an improper integral.
(b) Show that $$\int _ { \frac { 4 } { 3 } } ^ { \infty } \frac { 1 } { 9 x ^ { 2 } + 16 } d x = k \pi$$ where \(k\) is a constant to be determined.

14 The function f is defined by $$f(x) = \frac{1}{4x^2 + 16x + 19} \quad (x \in \mathbb{R})$$ 14

1. Show, without using calculus, that the graph of \(y = f(x)\) has a stationary point at \(\left(-2, \frac{1}{3}\right)\)
   [3 marks] 14
2. Show that \(\int_{-2}^{-\frac{1}{2}} f(x) \, dx = \frac{\pi\sqrt{3}}{18}\)
   [5 marks] 14
3. Find the value of \(\int_{-2}^{\infty} f(x) \, dx\)
   Fully justify your answer.
   [2 marks]