First slide

Methods of integration

Question

$Let α \in (0, \frac{π}{2}) be fixed. If \int \frac{\sin (x + α)}{\sin (x - α)} d x = f_{1} (x) \cos 2 α + f_{2} (x) \sin 2 α + c where c constant of integration then$

Moderate

A

$f_{1} (x) = x - α, f_{2} (x) = \log_{e} |\sin (x + α)|$
B

$f_{1} (x) = x + α, f_{2} (x) = \log_{e} |\sin (x - α)|$
C

$f_{1} (x) = x - α, f_{2} (x) = \log_{e} |\sin (x - α)|$
D

$f_{1} (x) = x + α, f_{2} (x) = \log_{e} |\sin (x + α)|$

Solution

$\begin{array}{l} \int \frac{\sin (x + α)}{\sin (x - α)} d x = \int \frac{\sin (x - α + 2 α)}{\sin (x - α)} d x Put x - α = θ \Rightarrow d x = d θ \\ = \int \frac{\sin (θ + 2 α)}{\sin θ} d θ = \cos 2 α \int d θ + \sin 2 α \int \frac{\cos θ}{\sin θ} d θ = \cos 2 α \cdot (x - α) + \sin 2 α \log | \sin (x - α) | + c \end{array}$

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