Calculate The Following Integral Z Cos2 X Dx

How to Calculate the Integral

The integral of z cos²x dx involves integrating the product of a constant z and the squared cosine function. This type of integral is common in physics, engineering, and signal processing applications.

Step-by-Step Calculation

1. Identify the constant z and the trigonometric function cos²x.
2. Recall the trigonometric identity: cos²x = (1 + cos(2x))/2.
3. Substitute the identity into the integral: ∫ z cos²x dx = ∫ z (1 + cos(2x))/2 dx.
4. Distribute z/2: ∫ z/2 (1 + cos(2x)) dx.
5. Split the integral: ∫ z/2 dx + ∫ z/2 cos(2x) dx.
6. Integrate each term separately:
   • ∫ z/2 dx = (z/2)x + C₁
   • ∫ z/2 cos(2x) dx = (z/4) sin(2x) + C₂
7. Combine the results: (z/2)x + (z/4) sin(2x) + C, where C = C₁ + C₂.

The Formula

For definite integrals with limits a and b:

Worked Example

Let's calculate the integral of 3 cos²x dx from 0 to π.

1. Apply the formula: ∫[0,π] 3 cos²x dx = [(3/2)x + (3/4) sin(2x)] evaluated from 0 to π.
2. Evaluate at upper limit (π):
   • (3/2)π + (3/4) sin(2π) = (3/2)π + 0 = (3/2)π
3. Evaluate at lower limit (0):
   • (3/2)(0) + (3/4) sin(0) = 0 + 0 = 0
4. Subtract lower from upper: (3/2)π - 0 = (3/2)π ≈ 4.7124.

The result is (3/2)π, which is approximately 4.7124.

Interpreting the Result

The result of the integral represents the area under the curve of z cos²x between the specified limits. For the example above, the area under 3 cos²x from 0 to π is (3/2)π.

Key points to consider:

• The result depends on the constant z and the limits of integration.
• The sine term (sin(2x)) oscillates between -1 and 1, so its contribution to the integral depends on the limits.
• For odd multiples of π, sin(2x) will be zero, simplifying the result.