How to Find the Integral of cos²(x)dx with Trigonometric Identity and Simplification

∫cos²(x)dx

To find the integral of cos²(x)dx, we can use the trigonometric identity:

cos²(x) = (1/2)(1 + cos(2x))

Using this identity, the integral becomes:

∫cos²(x)dx = ∫(1/2)(1 + cos(2x))dx

Now, we can split the integral into two separate parts:

∫(1/2)dx + ∫(1/2)cos(2x)dx

The first integral is simply a constant term, so let’s solve that first:

(1/2)∫dx

The integral of dx is simply x, so the first part becomes:

(1/2)x

Now, let’s solve the second integral:

(1/2) ∫cos(2x)dx

We can simplify this further by using the substitution u = 2x, which implies du = 2dx:

(1/4)∫cos(u)du

The integral of cos(u)du is sin(u):

(1/4)sin(u)

Substituting back for u:

(1/4)sin(2x)

Therefore, the fully simplified integral of cos²(x)dx is:

∫cos²(x)dx = (1/2)x + (1/4)sin(2x) + C

where C is the constant of integration

To find the integral of cos²(x)dx, we can use the trigonometric identity:

cos²(x) = (1/2)(1 + cos(2x))

Using this identity, the integral becomes:

∫cos²(x)dx = ∫(1/2)(1 + cos(2x))dx

Now, we can split the integral into two separate parts:

∫(1/2)dx + ∫(1/2)cos(2x)dx

The first integral is simply a constant term, so let’s solve that first:

(1/2)∫dx

The integral of dx is simply x, so the first part becomes:

(1/2)x

Now, let’s solve the second integral:

(1/2) ∫cos(2x)dx

We can simplify this further by using the substitution u = 2x, which implies du = 2dx:

(1/4)∫cos(u)du

The integral of cos(u)du is sin(u):

(1/4)sin(u)

Substituting back for u:

(1/4)sin(2x)

Therefore, the fully simplified integral of cos²(x)dx is:

∫cos²(x)dx = (1/2)x + (1/4)sin(2x) + C

where C is the constant of integration.

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