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\( \frac{h-\omega}{d 3 \cdot} \int \frac{\left(\sin ^{2} \theta-\sin \theta\right) \cos \theta}{\sin ^{2} \theta+4 \sin \theta-8} d \theta \)

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\(\int \frac{(sin^2\theta - sin\theta)cos\theta}{sin^2\theta + 4sin\theta} d\theta\)

Let \( sin\theta = t\)

\(cos\theta \, dx = dt\)

\(= \int \frac{t^2 - t}{t^2 + 4t - 8} dt\)

\(= \int \left(1 + \frac{-5t + 8}{t^2 + 4t - 8}\right) dt\)

\(= t + \int\frac {\frac{-5}2 (2t + 4)+18}{t^2 + 4t -8} dt\)

\(= t - \frac52 \int \frac {2t + 4}{t^2 + 4t - 8}+ 18\int\frac{dt}{(t + 2)^2 - (\sqrt{12})^2}\)

\(= t - \frac52 log |t^2 + 4t -8| + \frac{18}{2\sqrt {12}} ln\left(\frac{(t -2) - \sqrt {12}}{(t -2) + \sqrt{12}}\right) + C\)

\(= sin\theta - \frac 52 log |sin^2\theta + 4sin\theta - 8| + \frac 9{2\sqrt 3} ln \left(\frac{sin\theta - 2-2\sqrt 3}{sin\theta - 2+ 2\sqrt 3}\right) + C\)

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