Integration with algebraic manipulation

7. (a) Show that \(\frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x }\) can be written as \(9 x ^ { - \frac { 1 } { 2 } } - 6 + x ^ { \frac { 1 } { 2 } }\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , x > 0\), and that \(y = \frac { 2 } { 3 }\) at \(x = 1\),
(b) find \(y\) in terms of \(x\).

3

1. Find \(\int ( 2 x + 1 ) ( x + 3 ) \mathrm { d } x\).
2. Evaluate \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).

2

1. Write down the value of \(n\) given that \(\frac { 1 } { x ^ { 4 } } = x ^ { n }\).
2. Expand \(\left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 }\).
3. Hence find \(\int \left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 } \mathrm {~d} x\).
4. Hence find the exact value of \(\int _ { 1 } ^ { 3 } \left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 } \mathrm {~d} x\).

3

1. \(\quad\) Expand \(\left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 }\).
2. Hence find \(\int \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).
3. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).

2

1. Find \(\int \left( 1 + 3 x ^ { \frac { 1 } { 2 } } + x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
2. 1. The expression \(( 1 + y ) ^ { 3 }\) can be written in the form \(1 + 3 y + n y ^ { 2 } + y ^ { 3 }\). Write down the value of the constant \(n\).
   2. Hence, or otherwise, expand \(( 1 + \sqrt { x } ) ^ { 3 }\).
3. Hence find the exact value of \(\int _ { 0 } ^ { 1 } ( 1 + \sqrt { x } ) ^ { 3 } \mathrm {~d} x\).