Integration using inverse trig and hyperbolic functions
15
Given that
$$y = \operatorname { cosec } \theta$$
15
Express \(y\) in terms of \(\sin \theta\).
15
(ii) Hence, prove that
$$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta$$
15
(iii) Show that
$$\frac { \sqrt { y ^ { 2 } - 1 } } { y } = \cos \theta \quad \text { for } 0 < \theta < \frac { \pi } { 2 }$$
15
Use the substitution
$$x = 2 \operatorname { cosec } u$$
to show that
$$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x \quad \text { for } x > 2$$
can be written as
$$k \int \sin u \mathrm {~d} u$$
where \(k\) is a constant to be found.
15
(ii) Hence, show
$$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x = \frac { \sqrt { x ^ { 2 } - 4 } } { 4 x } + c \quad \text { for } x > 2$$
where \(c\) is a constant.
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