∫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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