Direct substitution integration

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. 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 \(\int \frac { 10 } { ( 2 x - 7 ) ^ { 2 } } \mathrm {~d} x\).

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

6 Use the substitution \(u = 1 + \ln x\) to find \(\int \frac { \ln x } { x ( 1 + \ln x ) ^ { 2 } } \mathrm {~d} x\).

6 Use the substitution \(u = x ^ { 2 } - 2\) to find \(\int \frac { 6 x ^ { 3 } + 4 x } { \sqrt { x ^ { 2 } - 2 } } \mathrm {~d} x\).

1 Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to find \(\int \frac { 1 } { 1 + \sin x + \cos x } \mathrm {~d} x\).

1

1. Differentiate the following.
   1. \(\frac { x ^ { 2 } } { 2 x + 1 }\)
   2. \(\tan \left( x ^ { 2 } - 3 x \right)\)
2. Use the substitution \(u = \sqrt { x } - 1\) to integrate \(\frac { 1 } { \sqrt { x } - 1 }\).
3. Integrate \(\frac { x - 2 } { 2 x ^ { 2 } - 8 x - 1 }\).

16 In this question you must show detailed reasoning.
Find \(\int \frac { x } { 1 + \sqrt { x } } d x\). END OF QUESTION PAPER

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 \(x\).

3. (a) Use the substitution \(u = 2 - x ^ { 2 }\) to find $$\int \frac { x } { 2 - x ^ { 2 } } \mathrm {~d} x$$ (b) Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 3 x \cos x d x$$

1. continued

\begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = x \sqrt { \ln x } , x \geq 1\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).
(a) Using the trapezium rule with two intervals of equal width, estimate the area of the shaded region. The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(b) Find the exact volume of the solid formed.

7

1. Find \(\int 5 x ^ { 3 } \sqrt { x ^ { 2 } + 1 } \mathrm {~d} x\).
2. Find \(\int \theta \tan ^ { 2 } \theta \mathrm {~d} \theta\). You may use the result \(\int \tan \theta \mathrm { d } \theta = \ln | \sec \theta | + c\).