First slide

Methods of integration

Question

If the primitive of $\sin^{- 3 / 2} x \sin^{- 1 / 2} (x + θ)$
is $- 2 cosec θ \sqrt{f (x)} + C$ then

Moderate

A

$f (x) = \frac{\sin x}{\sin (x + θ)}$
B

$f (x) = \tan (x + θ)$
C

$f (x) = \frac{\sin (x + θ)}{\sin x}$
D

$f (x) = \frac{\tan (x + θ)}{\tan x}$

Solution

The primitive of the given function is
$\begin{array}{l} \int \frac{d x}{\sqrt{\sin^{3} x \sin (x + θ)}} \\ = \int \frac{d x}{\sqrt{\sin^{3} x (\sin x \cos θ + \cos x ∣ \sin θ)}} \\ = \int \frac{{cosec}^{2} x d x}{\sqrt{\cos θ + \cot x \sin θ}} \\ = - \frac{1}{\sqrt{\sin θ}} \int \frac{d t}{\sqrt{\cot θ + t}} (t = \cot x) \\ = - \frac{2}{\sqrt{\sin θ}} (\cot θ + t)^{1 / 2} + C \\ = - \frac{2}{\sin θ} (\cos θ + \sin θ t)^{1 / 2} + C \\ = \frac{- 2 cosec θ (\sin x \cos θ + \sin θ \cos x)^{1 / 2} + C}{\sqrt{\sin x}} \end{array}$
$= - 2 cosec θ {(\frac{\sin (θ + x)}{\sin x})}^{1 / 2} + C .$

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