Question

\[\int\frac{\left( 3 \sin x - 2 \right) \cos x}{5 - \cos^2 x - 4 \sin x} dx\]

Answer 1

` ∫   {( 3 sin x -2 ) cos x   dx}/{5 cos^2 x -4 sin x} `
` ∫   {( 3 sin x -2 ) cos x   dx}/{5 -(1 - sin^2)-4 sin x} `
` ∫   {( 3 sin x -2 ) cos x   dx}/{sin^2 x -4  sin x + 4}`

`\text{ Let sin x }= t`
\[ \Rightarrow \text{ cos x dx} = dt\]
\[\int\frac{\left( 3t - 2 \right) dt}{t^2 - 4t + 4}\]
\[3t - 2 = A\frac{d}{dx}\left( t^2 - 4t + 4 \right) + B\]
\[3t - 2 = A \left( 2t - 4 \right) + B\]
\[3t - 2 = \left( 2 A \right) t + B - 4 A\]

Comparing the Coefficients of like powers of t

\[\text{ 2 A }= 3\]
\[A = \frac{3}{2}\]
\[\text{ B - 4  A }= - 2\]
\[B - 4 \times \frac{3}{2} = - 2\]
\[B = - 2 + 6\]
\[B = 4\]

\[3t - 2 = \frac{3}{2} \left( 2t - 4 \right) + 4\]
\[ \therefore \int\frac{\left( 3t - 2 \right) dt}{t^2 - 4t + 4}\]
\[ = \int\left( \frac{\frac{3}{2}\left( 2t - 4 \right) + 4}{t^2 - 4t + 4} \right)dt\]
\[ = \frac{3}{2}\int\left( \frac{2t - 4}{t^2 - 4t + 4} \right)dt + 4\int\frac{dt}{t^2 - 4t + 4}\]
`  =  3/2    I_1 + 4      I_2  . . . ( 1 )`
 Where
\[ I_1 = \int\frac{\left( 2t - 4 \right) dt}{t^2 - 4t + 4}, I_2 = \int\frac{dt}{t^2 - 4t + 4}\]
\[ I_1 = \int\frac{\left( 2t - 4 \right) dt}{t^2 - 4t + 4}\]
\[\text{ Let t }^2 - 4t + 4 = p\]
\[ \Rightarrow \left( 2t - 4 \right) dt = dp\]
\[ I_1 = \int\frac{\left( 2t - 4 \right) dt}{t^2 - 4t + 4}\]
\[ = \int\frac{dp}{p}\]
\[ = \text{ log }\left| p \right| + C_1 \]
\[ = \text{ log }\left| t^2 - 4t + 4 \right| + C_1 . . . \left( 2 \right)\]
\[ I_2 = \int\frac{dt}{t^2 - 4t + 4}\]
\[ I_2 = \int\frac{dt}{\left( t - 2 \right)^2}\]
\[ I_2 = \int \left( t - 2 \right)^{- 2} dt\]
\[ I_2 = \frac{\left( t - 2 \right)^{- 2 + 1}}{- 2 + 1} + C_2 \]
\[ I_2 = \frac{- 1}{t - 2} + C_2 . . . \left( 3 \right)\]
\[\text{ from }\left( 1 \right), \left( 2 \right) \text{ and }\left( 3 \right)\]
` ∫   {( 3 sin x -2 ) cos x   dx}/{5 cos^2 x -4 sin x} `
\[ = \frac{3}{2} \text{ log } \left| t^2 - 4t + 4 \right| + 4 \times \frac{- 1}{t - 2} + C_1 + C_2 \]
\[ = \frac{3}{2} \text{ log }\left| \sin^2 x - 4 \sin x + 4 \right| + \frac{4}{2 - t} + C \left( \text{ Where C }= C_1 + C_2 \right)\]
\[ = \frac{3}{2}\text{ log }\left| \left( \sin x - 2 \right)^2 \right| + \frac{4}{2 - \sin x} + C\]
\[ = \frac{3}{2} \times 2 \text{ log }\left| \sin x - 2 \right| + \frac{4}{2 - \sin x} + C\]
\[ = 3 \text{ log }\left| 2 - \sin x \right| + \frac{4}{2 - \sin x} + C\]