Choose the correct option from given four options: dx (x-a) (x-b) is equal to: 1. (b-a) | (x-b) (x-a) |+C 2. cosec(b-a) | (x-a) (x-b) |+C 3. cosec(b-a) | (x-b) (x-a) |+C 4. (b-a) | (x-b) (x-a) |+C

3. cosec(b-a) | (x-b) (x-a) |+C Solution: Let I= dx (x- a) (x-b) =1 (b-a) (b-a) (x- a) (x-b) dx =1 (b-a) (x-a-x+b) (x- a) (x-b) dx =1 (b-a) (x-a)-(x+b) (x- a) (x-b) dx =1 (b-a) (x-a) (x-b)- (x-a) (x-b) (x- a) (x-b) dx =1 (b-a) (x-a) (x-b) (x- a) (x-b) - (x-a) (x-b) (x-a ) (x-b) dx =1 (b-a) (x-b) (x- a) - (x-a) (x-a ) dx =1 (b-a) [ |(x-b)|- | (x-b)| ]+C =cosec(b-a) | (x-b) (x-a) |+C

Choose the correct option from given four options: dx (x-a) (x-b)…

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Question

Choose the correct option from given four options:
$\frac{\text{dx}}{\sin(\text{x}-\text{a})\sin(\text{x}-\text{b})}$ is equal to:

$\sin(\text{b}-\text{a)}\log\Big|\frac{\sin(\text{x}-\text{b})}{\sin(\text{x}-\text{a})}\Big|+\text{C}$
$\text{cosec}(\text{b}-\text{a)}\log\Big|\frac{\sin(\text{x}-\text{a})}{\sin(\text{x}-\text{b})}\Big|+\text{C}$
$\text{cosec}(\text{b}-\text{a)}\log\Big|\frac{\sin(\text{x}-\text{b})}{\sin(\text{x}-\text{a})}\Big|+\text{C}$
$\sin(\text{b}-\text{a)}\log\Big|\frac{\sin(\text{x}-\text{b})}{\sin(\text{x}-\text{a})}\Big|+\text{C}$

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Answer

$\text{cosec}(\text{b}-\text{a)}\log\Big|\frac{\sin(\text{x}-\text{b})}{\sin(\text{x}-\text{a})}\Big|+\text{C}$

Solution:
Let $\text{I}=\frac{\text{dx}}{\sin(\text{x}-\text{a})\sin(\text{x}-\text{b})}$
$=\frac{1}{\sin(\text{b}-\text{a})}\int\frac{\sin(\text{b}-\text{a})}{\sin(\text{x}-\text{a})\sin(\text{x}-\text{b})}\text{dx}$
$=\frac{1}{\sin(\text{b}-\text{a})}\int\frac{\sin(\text{x}-\text{a}-\text{x}+\text{b})}{\sin(\text{x}-\text{a})\sin(\text{x}-\text{b})}\text{dx}$
$=\frac{1}{\sin(\text{b}-\text{a})}\int\frac{\sin\big\{(\text{x}-\text{a})-(\text{x}+\text{b})\big\}}{\sin(\text{x}-\text{a})\sin(\text{x}-\text{b})}\text{dx}$
$=\frac{1}{\sin(\text{b}-\text{a})}\int\frac{\sin(\text{x}-\text{a})\cos(\text{x}-\text{b})-\cos(\text{x}-\text{a})\sin(\text{x}-\text{b})}{\sin(\text{x}-\text{a})\sin(\text{x}-\text{b})}\text{dx}$
$=\frac{1}{\sin(\text{b}-\text{a})}\int\frac{\sin(\text{x}-\text{a})\cos(\text{x}-\text{b})}{\sin(\text{x}-\text{a})\sin(\text{x}-\text{b})}-\frac{\cos(\text{x}-\text{a})\sin(\text{x}-\text{b})}{\sin(\text{x}-\text{a} )\sin(\text{x}-\text{b})}\text{dx}$
$=\frac{1}{\sin(\text{b}-\text{a})}\int\frac{\cos(\text{x}-\text{b})}{\sin(\text{x}-\text{a})}-\frac{\cos(\text{x}-\text{a})}{\sin(\text{x}-\text{a} )}\text{dx}$
$=\frac{1}{\sin(\text{b}-\text{a})}\big[\log\sin|(\text{x}-\text{b})|-\log|\sin(\text{x}-\text{b})|\big]+\text{C}$
$=\text{cosec}(\text{b}-\text{a})\log\Big|\frac{\sin(\text{x}-\text{b})}{\sin(\text{x}-\text{a})}\Big|+\text{C}$