Find: (3 x-2) x 13 - ^2x - 7 x dx - Vidyadip

Question

$\text{Find}:\int\frac{(3\sin x-2)\cos x}{13\ -\ \cos^2x \ - \ 7\sin x}\text{d}x$

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Answer

$\int\frac{(3\sin\text{x}-2)\cos\text{x}}{13\ -\ \cos^2\text{x}\ -\ 7\sin\text{x}}\text{dx}$
$\int\frac{(3\sin\text{x}-2)\cos\text{x}}{\sin^2\text{x}\ -\ 7\sin\text{x}\ +12}\text{dx}$
put sin x = y, cos x dx = dy
$=\int\frac{(3\text{y}-2)\text{dy}}{\text{y}^2-7\text{y}+12}$
$=\int\frac{(3\text{y}-2)\text{dy}}{(\text{y}-4)(\text{y}-3)}$
$=\int\Big(\frac{10}{\text{y}-4}-\frac{7}{\text{y}-3}\Big)\text{dy}$
= 10 log | y – 4 | –7 log | y – 3 | + C
= 10 log | sin x – 4 | – 7 log | sin x – 3 | + C

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