Integral Of Cos 2 X

The integral of cos(2x) is a fundamental trigonometric integral that can be solved using various techniques. To solve this integral, we can use the following approach:

∫cos(2x) dx = ∫cos^2(x) - sin^2(x) dx (using the double angle formula for cosine)

Now, we can use the trigonometric identities:

cos^2(x) = (1 + cos(2x))/2 sin^2(x) = (1 - cos(2x))/2

Substituting these identities into the integral, we get:

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

This may seem like a circular argument, but we can actually use a different approach to solve the integral.

Let’s use the substitution:

u = 2x du = 2dx

Then, we can rewrite the integral as:

∫cos(u) du/2 = (¹⁄₂) ∫cos(u) du

The integral of cos(u) is:

∫cos(u) du = sin(u) + C

Substituting back u = 2x, we get:

(¹⁄₂) ∫cos(u) du = (¹⁄₂) sin(2x) + C

So, the final answer is:

∫cos(2x) dx = (¹⁄₂) sin(2x) + C

Where C is the constant of integration.

Example Use Cases

1. Physics: The integral of cos(2x) is used to describe the motion of a pendulum. The equation of motion for a simple pendulum is given by:

d^2θ/dt^2 + (g/L) * sin(θ) = 0

where θ is the angle of the pendulum from the vertical, g is the acceleration due to gravity, and L is the length of the pendulum.

Using the small angle approximation, sin(θ) ≈ θ, and the equation of motion becomes:

d^2θ/dt^2 + (g/L) * θ = 0

The solution to this equation involves the integral of cos(2x), which describes the oscillatory motion of the pendulum.

1. Engineering: The integral of cos(2x) is used in the design of electronic filters. For example, a band-pass filter can be designed using a combination of resistors, capacitors, and inductors. The transfer function of the filter is given by:

H(ω) = 1 / (1 + jωRC)

where ω is the frequency, R is the resistance, C is the capacitance, and j is the imaginary unit.

The integral of cos(2x) is used to calculate the phase shift of the filter, which is given by:

φ(ω) = arctan(ωRC)

The phase shift is an important parameter in filter design, as it determines the timing of the output signal.

Code Implementation

Here is an example implementation of the integral of cos(2x) in Python:

import numpy as np
from scipy.integrate import quad

def integrand(x):
    return np.cos(2*x)

def integral_cos_2x():
    x = np.linspace(0, np.pi, 100)
    y = integrand(x)
    integral, _ = quad(integrand, 0, np.pi)
    return integral

result = integral_cos_2x()
print(result)

This code defines the integrand function integrand(x) = cos(2x) and uses the quad function from the scipy.integrate module to calculate the definite integral from 0 to π. The result is printed to the console.

FAQs

1. What is the integral of cos(2x) + sin(2x)?

The integral of cos(2x) + sin(2x) can be solved using the same techniques as above. The answer is:

∫(cos(2x) + sin(2x)) dx = (¹⁄₂) sin(2x) - (¹⁄₂) cos(2x) + C

1. What is the integral of cos^2(2x)?

The integral of cos^2(2x) can be solved using the trigonometric identity:

cos^2(2x) = (1 + cos(4x))/2

Then, we can integrate:

∫cos^2(2x) dx = ∫((1 + cos(4x))/2) dx = (¹⁄₂) ∫dx + (¹⁄₂) ∫cos(4x) dx = (¹⁄₂) x + (¹⁄₈) sin(4x) + C

1. What is the integral of cos(2x) from 0 to π?

The definite integral of cos(2x) from 0 to π can be calculated using the quad function in Python:

result, _ = quad(integrand, 0, np.pi)
print(result)

The result is approximately 0.0, since the integral of cos(2x) over a full period is zero.