Indefinite integral with linear substitution

8. (i) Find, using algebraic integration, the exact value of $$\int _ { 3 } ^ { 42 } \frac { 2 } { 3 x - 1 } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) $$\mathrm { h } ( x ) = \frac { 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 1 } { ( x - 1 ) ^ { 2 } } \quad x > 1$$ Given \(\mathrm { h } ( x ) = A x + B + \frac { C } { ( x - 1 ) ^ { 2 } }\) where \(A , B\) and \(C\) are constants to be found, find $$\int \mathrm { h } ( x ) \mathrm { d } x$$ \includegraphics[max width=\textwidth, alt={}, center]{1c700103-ecab-4a08-b411-3f445ed88885-26_2258_47_312_1985}

1. (i) Find

$$\int \frac { 12 } { ( 2 x - 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) (a) Write \(\frac { 4 x + 3 } { x + 2 }\) in the form $$A + \frac { B } { x + 2 } \text { where } A \text { and } B \text { are constants to be found }$$ (b) Hence find, using algebraic integration, the exact value of $$\int _ { - 8 } ^ { - 5 } \frac { 4 x + 3 } { x + 2 } d x$$ giving your answer in simplest form.

4. (i) Find $$\int _ { 5 } ^ { 13 } \frac { 1 } { ( 2 x - 1 ) } \mathrm { d } x$$ writing your answer in its simplest form.
(ii) Use integration to find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin 2 x + \sec \frac { 1 } { 3 } x \tan \frac { 1 } { 3 } x \mathrm {~d} x$$

4. Find

1. \(\int ( 2 x + 3 ) ^ { 12 } \mathrm {~d} x\)
2. \(\int \frac { 5 x } { 4 x ^ { 2 } + 1 } \mathrm {~d} x\)

1 Find \(\int \sqrt [ 3 ] { 2 x - 1 } \mathrm {~d} x\).

1 Find

1. \(\int 8 \mathrm { e } ^ { - 2 x } \mathrm {~d} x\),
2. \(\int ( 4 x + 5 ) ^ { 6 } \mathrm {~d} x\).

1 Find

1. \(\int 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x\),
2. \(\int 10 ( 2 x + 1 ) ^ { - 1 } \mathrm {~d} x\).

1 Find

1. \(\quad \int ( 4 - 3 x ) ^ { 7 } \mathrm {~d} x\),
2. \(\quad \int ( 4 - 3 x ) ^ { - 1 } \mathrm {~d} x\).

2 Find

1. \(\int \left( 2 - \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\),
2. \(\int ( 4 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).

2 Find \(\int \sqrt [ 3 ] { 2 x - 1 } \mathrm {~d} x\).