Improper integral to infinity

10 The function f is defined by \(\mathrm { f } ( x ) = ( 4 x + 2 ) ^ { - 2 }\) for \(x > - \frac { 1 } { 2 }\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
   A point is moving along the curve \(y = \mathrm { f } ( x )\) in such a way that, as it passes through the point \(A\), its \(y\)-coordinate is decreasing at the rate of \(k\) units per second and its \(x\)-coordinate is increasing at the rate of \(k\) units per second.
2. Find the coordinates of \(A\).

1 Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 2 } { x ^ { 2 } } d x\).

2

1. Find \(\int _ { 0 } ^ { a } \left( \mathrm { e } ^ { - x } + 6 \mathrm { e } ^ { - 3 x } \right) \mathrm { d } x\), where \(a\) is a positive constant.
2. Deduce the value of \(\int _ { 0 } ^ { \infty } \left( \mathrm { e } ^ { - x } + 6 \mathrm { e } ^ { - 3 x } \right) \mathrm { d } x\).

1. (a) Evaluate the integral

$$\int _ { 0 } ^ { \infty } \frac { \mathrm { d } x } { ( 1 + x ) ^ { 5 } }$$ (b) By putting \(u = \ln x\), determine whether or not the following integral has a finite value. $$\int _ { 2 } ^ { \infty } \frac { \mathrm { d } x } { x \ln x }$$