Definite integral with simple linear/polynomial substitution

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

5. (i) Find, by algebraic integration, the exact value of $$\int _ { 2 } ^ { 4 } \frac { 8 } { ( 2 x - 3 ) ^ { 3 } } d x$$ (ii) Find, in simplest form, $$\int x \left( x ^ { 2 } + 3 \right) ^ { 7 } d x$$

1. Evaluate

$$\int _ { 2 } ^ { 15 } \frac { 1 } { \sqrt [ 3 ] { 2 x - 3 } } d x$$

6

1. Find \(\int ( 2 x - 3 ) ^ { 7 } \mathrm {~d} x\).
2. Use the substitution \(u = x ^ { 2 } + 1\), or otherwise, to find \(\int _ { 1 } ^ { 2 } x \left( x ^ { 2 } + 1 \right) ^ { 3 } \mathrm {~d} x\).

4 Evaluate the following integrals, giving your answers in exact form. \begin{displayquote}

1. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x ^ { 2 } + 1 } \mathrm {~d} x\)
2. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x + 1 } \mathrm {~d} x\) \end{displayquote}

6. (i) Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
(ii) Using the substitution \(u = x ^ { 2 } + 4\), evaluate $$\int _ { 0 } ^ { 2 } \frac { 5 x } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } d x$$

4 Find the exact value of \(\int _ { 0 } ^ { 2 } \sqrt { 1 + 4 x } \mathrm {~d} x\), showing your working.

4 Evaluate the following integrals, giving your answers in exact form.

1. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x ^ { 2 } + 1 } \mathrm {~d} x\).
2. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x + 1 } \mathrm {~d} x\).

4 Use the substitution \(u = 2 + \ln t\) to find the exact value of $$\int _ { 1 } ^ { \mathrm { e } } \frac { 1 } { t ( 2 + \ln t ) ^ { 2 } } \mathrm {~d} t$$

1. (i) Use the substitution \(t = \tan \frac { X } { 2 }\) to prove the identity

$$\frac { \sin x - \cos x + 1 } { \sin x + \cos x - 1 } \equiv \sec x + \tan x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$ (ii) Use the substitution \(t = \tan \frac { \theta } { 2 }\) to determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { 5 } { 4 + 2 \cos \theta } d \theta$$ giving your answer in simplest form.

Use the substitution \(u = 4 + 3x^2\) to find the exact value of $$\int_0^2 \frac{2x}{(4 + 3x^2)^2} \, dx .$$ [6]

Evaluate $$\int_2^6 \sqrt{3x-2} \, dx.$$ [4]

Find the exact value of \(\int_0^2 \sqrt{1+4x} \, dx\), showing your working. [5]

Use the substitution \(u = 1 + \sin x\) to find the value of $$\int_0^{\frac{\pi}{4}} \cos x (1 + \sin x)^3 \, dx.$$ [6]