Find (3 - - 2) 5 - ^ 2 - 4 d . - Vidyadip

Question

$\text{Find}\int\frac{(3 - \sin\theta - 2)\cos\theta}{5 - \cos^{2}\theta - 4 \sin\theta} \text{d}\theta.$

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Answer

$\text{I} = \int\frac{(3\sin\theta - 2) \cos\theta}{5 - (1 -\sin^{2}\theta) - 4\sin\theta}\text{d}\theta$
$\sin\theta = \text{t} \Rightarrow \cos\theta\text{d}\theta = \text{dt}$
$\therefore\text{I} = \int\frac{3\text{t} - 2}{\text{t}^{2} - 4\text{t} + 4} = \int\frac{3\text{t} - 2}{(\text{t} - 2)^{2}}\text{dt}$
$= \int\frac{3(\text{t} - 2)}{(\text{t - 2)}^{2}}\text{dt} + 4 \int\frac{1}{(\text{t - 2)}^{2}}\text{dt}$
$= 3\log|\text{t} = 2| - \frac{4}{(\text{t - 2)}} + \text{C}$
$= 3\log|\sin\theta - 2| - \frac{4}{(\sin\theta - 2)} + \text{C}$

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