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lemonsisterd's Question
Posted about 2 months ago

8. Find, in simplest form,
\int(2 \cos x-\sin x)^{2} d x
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Answer from Sia
Posted about 2 months ago
Solution by Steps
step 1
We start by expanding the integrand (2cosxsinx)2(2 \cos x - \sin x)^2
step 2
Expanding (2cosxsinx)2(2 \cos x - \sin x)^2 gives 4cos2x4cosxsinx+sin2x4 \cos^2 x - 4 \cos x \sin x + \sin^2 x
step 3
We use trigonometric identities to simplify: cos2x=1+cos2x2\cos^2 x = \frac{1 + \cos 2x}{2} and sin2x=1cos2x2\sin^2 x = \frac{1 - \cos 2x}{2}
step 4
Substituting these identities, we get 4(1+cos2x2)4cosxsinx+(1cos2x2)4 \left(\frac{1 + \cos 2x}{2}\right) - 4 \cos x \sin x + \left(\frac{1 - \cos 2x}{2}\right)
step 5
Simplifying further, we obtain 2(1+cos2x)4cosxsinx+1cos2x22 (1 + \cos 2x) - 4 \cos x \sin x + \frac{1 - \cos 2x}{2}
step 6
Combining like terms, we get 52+32cos2x4cosxsinx\frac{5}{2} + \frac{3}{2} \cos 2x - 4 \cos x \sin x
step 7
We now integrate each term separately: 52dx+32cos2xdx4cosxsinxdx\int \frac{5}{2} dx + \int \frac{3}{2} \cos 2x dx - \int 4 \cos x \sin x dx
step 8
Integrating each term, we get 52x+34sin2x+cos2x+C\frac{5}{2} x + \frac{3}{4} \sin 2x + \cos 2x + C
52x+34sin2x+cos2x+C\frac{5}{2} x + \frac{3}{4} \sin 2x + \cos 2x + C
Key Concept
Trigonometric Identities and Integration
We used trigonometric identities to simplify the integrand and then integrated each term separately.

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