First slide

Introduction to integration

Question

$If \int \frac{1}{\sin (x - a) \sin (x - b)} d x = A \log |\frac{\sin (x - a)}{\sin (x - b)}| + c$

Easy

A

$\sin (a - b)$
B

Sin(b-a)
C

cosec(b-a)
D

cosec(a-b)

Solution

$\begin{array}{l} \int \frac{1}{\sin (x - a) \sin (x - b)} d x \\ = \frac{1}{\sin (a - b)} \int \frac{\sin {(x - b) - (x - a)}}{\sin (x - a) \sin (x - b)} d x \\ = \frac{1}{\sin (a - b)} \int \frac{\sin (x - b) \cos (x - a) - \cos (x - b) \sin (x - a)}{\sin (x - a) \sin (x - b)} d x \\ = \frac{1}{\sin (a - b)} \int {\cot (x - a) - \cot (x - b)} d x \\ = \frac{1}{\sin (a - b)} {\log | \sin (x - a) | - \log | \sin (x - b) |} + c \\ = cosec (a - b) \log |\frac{\sin (x - a)}{\sin (x - b)}| + c \end{array}$

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