How to Solve the Integral of cos²(x): Step-by-Step Guide with Trigonometric Identities

∫cos²(x)dx

To solve the integral ∫cos²(x)dx, we can use trigonometric identities to rewrite it in a more convenient form

To solve the integral ∫cos²(x)dx, we can use trigonometric identities to rewrite it in a more convenient form.

First, recall the double-angle identity for cosine:

cos(2θ) = cos²(θ) – sin²(θ)

Rearranging this equation, we can solve for cos²(θ):

cos²(θ) = (1/2)(1 + cos(2θ))

Now, let’s substitute 2θ with x in order to match the integral:

cos²(x) = (1/2)(1 + cos(2x))

Now, we can rewrite the integral as:

∫cos²(x)dx = ∫(1/2)(1 + cos(2x))dx

To calculate this integral, we use the linearity property of integrals:

∫(1/2)(1 + cos(2x))dx = (1/2)∫(1 + cos(2x))dx

The integral of 1 with respect to x is x, so we have:

(1/2)∫(1 + cos(2x))dx = (1/2)(∫1dx + ∫cos(2x)dx)

The integral of 1 with respect to x is simply x, so we have:

(1/2)(∫dx + ∫cos(2x)dx) = (1/2)(x + ∫cos(2x)dx)

To find the integral of cos(2x), we can use the substitution method. Let u = 2x, which implies du = 2dx:

(1/2)(x + ∫cos(u)(1/2)du)

Now, we integrate cos(u) with respect to u, which is sin(u):

(1/2)(x + (1/2)∫sin(u)du)

The integral of sin(u) with respect to u is -cos(u):

(1/2)(x – (1/2)cos(u))

Finally, we substitute back u = 2x:

(1/2)(x – (1/2)cos(2x)) + C

So, the solution to the integral ∫cos²(x)dx is (1/2)(x – (1/2)cos(2x)) + C, where C is the constant of integration.

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