Integral Of Cos2x

The integral of cos(2x) is a fundamental problem in calculus that involves integrating a trigonometric function. Let’s break down the process step by step to understand how to solve it.

Understanding the Problem

We are asked to find the integral of cos(2x) with respect to x. In mathematical notation, this is represented as:

∫cos(2x) dx

At first glance, this might seem like a straightforward integral, but the presence of the 2x inside the cosine function requires a specific approach.

Substitution Method

To solve this integral, we’ll use the substitution method, a powerful technique in calculus. Let’s set:

u = 2x

Now, we need to find du, the differential of u:

du = 2 dx

or equivalently,

dx = du/2

Rewriting the Integral

Substitute u and dx into the original integral:

∫cos(u) (du/2)

We can rewrite this as:

(¹⁄₂) ∫cos(u) du

Now, the integral is in terms of u, and we can proceed with the integration.

Integrating cos(u)

The integral of cos(u) with respect to u is a standard result in calculus:

∫cos(u) du = sin(u) + C

where C is the constant of integration.

Substituting Back

Now, substitute back u = 2x into the result:

(¹⁄₂) sin(u) + C/2 = (¹⁄₂) sin(2x) + C/2

To maintain the standard form of the constant of integration, we can rewrite C/2 as a new constant, C. However, it’s essential to note that this new constant is half the original constant. To avoid confusion, we’ll keep the original constant and adjust its representation:

(¹⁄₂) sin(2x) + C/2 = (¹⁄₂) sin(2x) + C

where the new C is half the original constant.

Final Answer

The integral of cos(2x) with respect to x is:

∫cos(2x) dx = (¹⁄₂) sin(2x) + C

where C is the constant of integration.

Verification

To verify our result, we can differentiate the obtained antiderivative with respect to x:

d/dx [(¹⁄₂) sin(2x) + C] = (¹⁄₂) ⋅ 2 ⋅ cos(2x) = cos(2x)

This confirms that our solution is correct.

Alternative Approach: Double-Angle Formula

Another way to approach this problem is by using the double-angle formula for cosine:

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

However, these formulas don’t simplify the integral directly. The substitution method remains the most efficient approach for this problem.

Applications

The integral of cos(2x) has various applications in mathematics, physics, and engineering. Some examples include:

1. Vibration Analysis: In mechanical engineering, the integral of cos(2x) can be used to analyze the vibration of systems with harmonic motion.
2. Electrical Circuits: In electrical engineering, this integral can be applied to analyze AC circuits with sinusoidal inputs.
3. Signal Processing: In signal processing, the integral of cos(2x) is used in Fourier analysis to decompose signals into their frequency components.

FAQ Section

Conclusion

In conclusion, the integral of cos(2x) is a fundamental problem in calculus that can be solved using the substitution method. The result, (¹⁄₂) sin(2x) + C, has various applications in mathematics, physics, and engineering. By understanding the step-by-step process and verifying our solution, we can confidently apply this result to real-world problems.