Calculate The Given Integral Cos 4 11x Sin3 11x Dx

Introduction

Calculating the integral of cos(4*11x) * sin(3*11x) dx involves applying trigonometric identities to simplify the integrand before performing the integration. This type of integral commonly appears in physics, engineering, and signal processing applications.

The integral represents the area under the curve of the product of two cosine and sine functions. The calculator provided on this page simplifies this process by handling the mathematical operations automatically.

Formula

The integral we're solving is:

To solve this, we can use the product-to-sum trigonometric identity:

Applying this identity to our integrand:

Now we can integrate term by term:

The integrals of sine functions are straightforward:

Therefore, the final result is:

Calculation Process

To calculate the integral using our calculator:

1. Enter the coefficient for the cosine function (4*11)
2. Enter the coefficient for the sine function (3*11)
3. Click "Calculate" to see the result
4. Review the step-by-step solution

The calculator performs the following steps automatically:

1. Applies the product-to-sum identity
2. Integrates each term separately
3. Combines the results with the constant of integration
4. Presents the final answer in a simplified form

Worked Example

Let's calculate the definite integral from 0 to π/2:

Using our formula:

Simplifying:

Since cos(37.65π) = cos(π) = -1:

The final result is approximately 0.012987 - 0.5 = -0.487013.

Interpreting Results

The result of the integral represents the net area between the curve and the x-axis over the interval. For definite integrals:

• A positive result indicates the curve is above the x-axis
• A negative result indicates the curve is below the x-axis
• The magnitude represents the area

In our example, the negative result suggests the curve spends more time below the x-axis than above within the interval [0, π/2].