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8. Find, in simplest form,

\int(2 \cos x-\sin x)^{2} d x

May 27, 2024

Solution by Steps

step 1

We start by expanding the integrand

(2 \cos x - \sin x)^2

step 2

Expanding

(2 \cos x - \sin x)^2

gives

4 \cos^2 x - 4 \cos x \sin x + \sin^2 x

step 3

We use trigonometric identities to simplify:

\cos^2 x = \frac{1 + \cos 2x}{2}

and

\sin^2 x = \frac{1 - \cos 2x}{2}

step 4

Substituting these identities, we get

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 + \cos 2x) - 4 \cos x \sin x + \frac{1 - \cos 2x}{2}

step 6

Combining like terms, we get

\frac{5}{2} + \frac{3}{2} \cos 2x - 4 \cos x \sin x

step 7

We now integrate each term separately:

\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

\frac{5}{2} x + \frac{3}{4} \sin 2x + \cos 2x + C

Answer

\frac{5}{2} x + \frac{3}{4} \sin 2x + \cos 2x + C

Key Concept

Trigonometric Identities and Integration

Explanation

We used trigonometric identities to simplify the integrand and then integrated each term separately.