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.

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

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]
2. Show that \(\int_{-2}^{-\frac{1}{2}} f(x) \, dx = \frac{\pi\sqrt{3}}{18}\) [5 marks]
3. Find the value of \(\int_{-2}^{\infty} f(x) \, dx\) Fully justify your answer. [2 marks]

In this question you must show detailed reasoning. Evaluate \(\int_0^{\infty} 2xe^{-x} dx\). [You may use the result \(\lim_{x \to \infty} xe^{-x} = 0\).] [4]

In this question you must show detailed reasoning. Find \(\int_{2}^{\infty} \frac{1}{4+x^2} \, dx\). [4]