Definite integral with complex substitution requiring algebraic rearrangement

5. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.
Using the substitution \(u = 3 + \sqrt { 2 x - 1 }\) find the exact value of $$\int _ { 1 } ^ { 13 } \frac { 4 } { 3 + \sqrt { 2 x - 1 } } d x$$ giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found.

3. Using the substitution \(u ^ { 2 } = 2 x - 1\), or otherwise, find the exact value of $$\int _ { 1 } ^ { 5 } \frac { 3 x } { \sqrt { ( 2 x - 1 ) } } \mathrm { d } x$$ (8)
(8)

3. Using the substitution \(u = 2 + \sqrt { } ( 2 x + 1 )\), or other suitable substitutions, find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { } ( 2 x + 1 ) } d x$$ giving your answer in the form \(A + 2 \ln B\), where \(A\) is an integer and \(B\) is a positive constant.

7. (i) Using a suitable substitution, find, using calculus, the value of $$\int _ { 1 } ^ { 5 } \frac { 3 x } { \sqrt { 2 x - 1 } } \mathrm {~d} x$$ (Solutions relying entirely on calculator technology are not acceptable.)
(ii) Find $$\int \frac { 6 x ^ { 2 } - 16 } { ( x + 1 ) ( 2 x - 3 ) } d x$$

3 Evaluate \(\int _ { 0 } ^ { 3 } x ( x + 1 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\), giving your answer as an exact fraction.

6 Use the substitution \(u = 2 x + 1\) to evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 4 x - 1 } { ( 2 x + 1 ) ^ { 5 } } \mathrm {~d} x\).

6

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 0.4 } \cos \sqrt { 3 x + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
2. Use the substitution \(u = 3 x + 1\) to find the exact value of \(\int _ { 0 } ^ { 1 } x \sqrt { 3 x + 1 } \mathrm {~d} x\).
   (6 marks)

2

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = ( 3 x - 1 ) ^ { 10 }\).
2. Use the substitution \(u = 2 x + 1\) to find \(\int x ( 2 x + 1 ) ^ { 8 } \mathrm {~d} x\), giving your answer in terms of \(x\).

6 Use the substitution \(u = x ^ { 4 } + 2\) to find the value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 7 } } { \left( x ^ { 4 } + 2 \right) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(p \ln q + r\), where \(p , q\) and \(r\) are rational numbers.

7 Use the substitution \(u = 3 - x ^ { 3 }\) to find the exact value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { 3 - x ^ { 3 } } \mathrm {~d} x\).
[0pt] [6 marks]

8 Use the substitution \(u = 4 x - 1\) to find the exact value of $$\int _ { \frac { 1 } { 4 } } ^ { \frac { 1 } { 2 } } ( 5 - 2 x ) ( 4 x - 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$$

7 Use the substitution \(u = 6 - x ^ { 2 }\) to find the value of \(\int _ { 1 } ^ { 2 } \frac { x ^ { 3 } } { \sqrt { 6 - x ^ { 2 } } } \mathrm {~d} x\), giving your answer in the form \(p \sqrt { 5 } + q \sqrt { 2 }\), where \(p\) and \(q\) are rational numbers.
[0pt] [7 marks]

a) Use a suitable substitution to find $$\int \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x$$ b) Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x\). END OF PAPER \end{document}

Evaluate \(\int_0^3 x(x + 1)^{-\frac{1}{2}} dx\), giving your answer as an exact fraction. [5]