Finding the Integral of cos²(x)dx | Step-by-Step Explanation and Solution

∫cos²(x)dx

To find the integral of cos²(x)dx, we can use the trigonometric identity for the double angle formula of cosine: cos(2x) = 2cos²(x) – 1

To find the integral of cos²(x)dx, we can use the trigonometric identity for the double angle formula of cosine: cos(2x) = 2cos²(x) – 1. Rearranging this identity, we have cos²(x) = (1/2)(1 + cos(2x)).

Now, let’s integrate using this identity:

∫cos²(x)dx = ∫(1/2)(1 + cos(2x))dx

Breaking this into two separate integrals:

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

The integral of 1 is simply x, so the first term becomes:

(1/2)∫1dx = (1/2)x + C1

For the second term, we can use the identity sin(2x) = 2sin(x)cos(x) to rewrite cos(2x) as sin(2x)/(2sin(x)):

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

Next, we can make a substitution to simplify this integral. Let u = sin(x), then du = cos(x)dx. Solving for dx, we have dx = du/cos(x).

Substituting these values into the integral, we get:

(1/2)∫(sin(2x)/(2sin(x)))dx = (1/2)∫(sin(2x)/(2u)) * (du/cos(x))

Notice that sin(2x) can be expressed as 2sin(x)cos(x):

(1/2)∫(2sin(x)cos(x)/(2u)) * (du/cos(x))

The cos(x) cancels out:

(1/2)∫(2sin(x)/(2u)) * du

Simplifying further:

(1/2)∫sin(x)/u du

Now, we can integrate this term:

(1/2)∫sin(x)/u du = (1/2)ln|u| + C2

Remembering that u = sin(x), we can substitute back:

(1/2)ln|u| + C2 = (1/2)ln|sin(x)| + C2

Now, combining our two terms, we have:

∫cos²(x)dx = (1/2)x + (1/2)ln|sin(x)| + C

Therefore, the integral of cos²(x)dx is equal to (1/2)x + (1/2)ln|sin(x)| + C, where C is the constant of integration.

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