Standard integral of 1/√(x²+a²)

1. (a) Determine

$$\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x$$ (b) Hence determine the exact value of $$\int _ { - 2 } ^ { 2 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } d x$$ Give your answer in the form \(a \ln ( b + c \sqrt { 13 } )\), where \(a , b\) and \(c\) are rational numbers.

2. (a) Find $$\int \frac { 1 } { \sqrt { } \left( 4 x ^ { 2 } + 9 \right) } d x$$ (b) Use your answer to part (a) to find the exact value of $$\int _ { - 3 } ^ { 3 } \frac { 1 } { \sqrt { \left( 4 x ^ { 2 } + 9 \right) } } d x$$ giving your answer in the form \(k \ln ( a + b \sqrt { } 5 )\), where \(a\) and \(b\) are integers and \(k\) is a constant.

1. The curve \(C\) has equation

$$y = \frac { 1 } { \sqrt { x ^ { 2 } + 2 x - 3 } } , \quad x > 1$$

1. Find \(\int y \mathrm {~d} x\) The region \(R\) is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 3\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find the volume of the solid generated. Give your answer in the form \(p \pi \ln q\), where \(p\) and \(q\) are rational numbers to be found.

6 By first completing the square, find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x ^ { 2 } + 4 x + 8 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.

6 Given that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 16 + 9 x ^ { 2 } } } \mathrm {~d} x + \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 9 + 4 x ^ { 2 } } } \mathrm {~d} x = \ln a$$ find the exact value of \(a\).

1 Find \(\int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x\), giving your answer exactly in logarithmic form.

3. $$f ( x ) = \frac { 1 } { \sqrt { 4 x ^ { 2 } + 9 } }$$

1. Using a substitution, that should be stated clearly, show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \sinh ^ { - 1 } ( B x ) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
2. Hence find, in exact form in terms of natural logarithms, the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\).

15 In this question you must show detailed reasoning. Show that \(\int _ { \frac { 3 } { 4 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 4 x + 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { 3 + \sqrt { 5 } } { 2 } \right)\).

16 In this question you must show detailed reasoning. Show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } } \mathrm { dx } = \ln \left( \frac { \mathrm { a } + \mathrm { b } \sqrt { 3 } } { \mathrm { c } } \right)\), where \(a , b\) and \(c\) are integers to be determined.

9 In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 11 } \frac { 1 } { \sqrt { x ^ { 2 } + 6 x + 13 } } \mathrm {~d} x\) giving your answer in the form \(\ln ( p + q \sqrt { 2 } )\) where \(p\) and \(q\) are integers to be determined.

\includegraphics{figure_12} Figure 1 shows the cross-section \(R\) of an artificial ski slope. The slope is modelled by the curve with equation $$y = \frac{10}{\sqrt{4x^2 + 9}}, \quad 0 \leq x \leq 5.$$ Given that 1 unit on each axis represents 10 metres, use integration to calculate the area \(R\). Show your method clearly and give your answer to 2 significant figures. [7]

Evaluate \(\int_1^4 \frac{1}{\sqrt{x^2 - 2x + 17}} \, dx\), giving your answer as an exact logarithm. [5]

Evaluate the integral $$\int_0^1 \frac{dx}{\sqrt{2x^2 + 4x + 6}}.$$ [6]

In this question you must show detailed reasoning. Find \(\int_{-1}^{11} \frac{1}{\sqrt{x^2 + 6x + 13}} dx\) giving your answer in the form \(\ln(p + q\sqrt{2})\) where \(p\) and \(q\) are integers to be determined. [7]