Completing square then standard inverse trig

1 Find the exact value of \(\int _ { 2 } ^ { \frac { 7 } { 2 } } \frac { 1 } { \sqrt { 4 x - x ^ { 2 } - 1 } } \mathrm {~d} x\).

5. Determine

1. \(\int \frac { 1 } { \sqrt { x ^ { 2 } - 3 x + 5 } } \mathrm {~d} x\)
2. \(\int \frac { 1 } { \sqrt { 63 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\)

3. Without using a calculator, find

1. \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\), giving your answer as a multiple of \(\pi\),
2. \(\int _ { - 1 } ^ { 4 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 34 } } \mathrm {~d} x\), giving your answer in the form \(p \ln ( q + r \sqrt { 2 } )\),
   where \(p , q\) and \(r\) are rational numbers to be found.
   

2. Determine

1. \(\int \frac { 1 } { 3 x ^ { 2 } + 12 x + 24 } \mathrm {~d} x\)
2. \(\int \frac { 1 } { \sqrt { 27 - 6 x - x ^ { 2 } } } \mathrm {~d} x\)

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Determine $$\int \frac { 1 } { \sqrt { 5 + 4 x - x ^ { 2 } } } d x$$
2. Use the substitution \(x = 3 \sec \theta\) to determine the exact value of $$\int _ { 2 \sqrt { 3 } } ^ { 6 } \frac { 18 } { \left( x ^ { 2 } - 9 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$ Give your answer in the form \(A + B \sqrt { 3 }\) where \(A\) and \(B\) are constants to be found.

2. $$9 x ^ { 2 } + 6 x + 5 \equiv a ( x + b ) ^ { 2 } + c$$

1. Find the values of the constants \(a\), \(b\) and \(c\). Hence, or otherwise, find
2. \(\int \frac { 1 } { 9 x ^ { 2 } + 6 x + 5 } d x\)
3. \(\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 6 x + 5 } } \mathrm {~d} x\)

1. (a) Write \(x ^ { 2 } + 4 x - 5\) in the form \(( x + p ) ^ { 2 } + q\) where \(p\) and \(q\) are integers.
   (b) Hence use a standard integral from the formula book to find

$$\int \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \mathrm {~d} x$$ (c) Determine the mean value of the function $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \quad 3 \leqslant x \leqslant 13$$ giving your answer in the form \(A \ln B\) where \(A\) and \(B\) are constants in simplest form.

1 By completing the square, or otherwise, find the exact value of \(\int _ { 2 } ^ { 6 } \frac { 1 } { x ^ { 2 } - 6 x + 12 } \mathrm {~d} x\).

Using calculus, find the exact values of

1. \(\int_1^2 \frac{1}{x^2 - 4x + 5} \, dx\) [3]
2. \(\int_{\sqrt{3}}^3 \frac{\sqrt{x^2 - 3}}{x^2} \, dx\) [5]

Without using a calculator, find

1. \(\int_{-2}^{1} \frac{1}{x^2 + 4x + 13} \, dx\), giving your answer as a multiple of \(\pi\), [5]
2. \(\int_{-1}^{4} \frac{1}{\sqrt{4x^2 - 12x + 34}} \, dx\), giving your answer in the form \(p \ln\left(q + r\sqrt{2}\right)\), [7]

where \(p\), \(q\) and \(r\) are rational numbers to be found.

Show that

1. \(\int_5^8 \frac{1}{x^2 - 10x + 34} dx = k\pi\), giving the value of the fraction \(k\), [5]
2. \(\int_5^8 \frac{1}{\sqrt{x^2 - 10x + 34}} dx = \ln(A + \sqrt{n})\), giving the values of the integers \(A\) and \(n\). [4]

Using calculus, find the exact value of

1. \(\int_1^2 \frac{1}{\sqrt{x^2 - 2x + 3}} \, dx\) [4]
2. \(\int_0^1 e^{-x} \sinh x \, dx\) [4]