Integrate after expanding or multiplying out

1 Find \(\int \left( x + \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\).

1. Find

$$\int ( 2 x - 5 ) ( 3 x + 2 ) ( 2 x + 5 ) \mathrm { d } x$$ writing your answer in simplest form.

4. Find $$\int \frac { ( 3 \sqrt { x } + 2 ) ( x - 5 ) } { 4 \sqrt { x } } d x$$ writing each term in simplest form.

6. (a) Show that \(( 4 + 3 \sqrt { } x ) ^ { 2 }\) can be written as \(16 + k \sqrt { } x + 9 x\), where \(k\) is a constant to be found.
(b) Find \(\int ( 4 + 3 \sqrt { } x ) ^ { 2 } \mathrm {~d} x\).

1. Find

$$\int \frac { x ^ { \frac { 1 } { 2 } } ( 2 x - 5 ) } { 3 } \mathrm {~d} x$$ writing each term in simplest form.

7

1. Find \(\int x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x\).
2. Show that \(\int _ { 0 } ^ { 8 } x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x = a \times 2 ^ { 11 }\), where \(a\) is a positive integer to be determined.