Showing integral equals given value

1. Show that

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin 2 \theta } { 1 + \cos \theta } d \theta = 2 - 2 \ln 2$$

3. Use the substitution \(x = \sin \theta\) to show that, for \(| x | \leq 1\), $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c \text {, where } c \text { is an arbitrary constant. }$$

2. Use the substitution \(x = 2 \tan u\) to show that $$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 1 } { 2 } ( 4 - \pi )$$

5. $$I = \int \frac { 1 } { 4 \cos x - 3 \sin x } \mathrm {~d} x \quad 0 < x < \frac { \pi } { 4 }$$ Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to show that $$I = \frac { 1 } { 5 } \ln \left( \frac { 2 + \tan \left( \frac { x } { 2 } \right) } { 1 - 2 \tan \left( \frac { x } { 2 } \right) } \right) + k$$ where \(k\) is an arbitrary constant.

5 By using a suitable substitution, which should be stated, show that $$\int _ { \frac { 3 } { 2 } } ^ { \frac { 5 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 13 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln ( 1 + \sqrt { 2 } )$$

5 Use integration by substitution to show that $$\int _ { - \frac { 1 } { 4 } } ^ { 6 } x \sqrt { 4 x + 1 } \mathrm {~d} x = \frac { 875 } { 12 }$$ Fully justify your answer.
[0pt] [6 marks]