Calculate The Given Integral Cos4 11x Sin3 11x Dx

This calculator helps you compute the definite or indefinite integral of cos(4*11x) * sin(3*11x) dx. Whether you're a student studying calculus or a professional working with trigonometric functions, this tool provides an accurate and efficient solution.

How to Calculate the Integral

Calculating the integral of cos(4*11x) * sin(3*11x) involves several steps. The process begins with identifying the trigonometric product and then applying appropriate identities to simplify the expression before integration.

Note: This integral requires the use of trigonometric identities to simplify the product of cosine and sine functions before integration.

Step 1: Identify the Trigonometric Product

The integral involves the product of cos(4*11x) and sin(3*11x). To simplify this, we can use the product-to-sum identities, which convert products of trigonometric functions into sums.

Step 2: Apply Product-to-Sum Identities

The product-to-sum identity for cosine and sine is:

cos(A) * sin(B) = [sin(A+B) - sin(A-B)] / 2

Applying this identity to our integral:

cos(4*11x) * sin(3*11x) = [sin(7*11x) - sin(x)] / 2

Step 3: Integrate the Simplified Expression

Now that the expression is simplified, we can integrate term by term:

∫ [sin(7*11x) - sin(x)] / 2 dx = (1/2) ∫ sin(7*11x) dx - (1/2) ∫ sin(x) dx

The integral of sin(kx) is -cos(kx)/k. Applying this to both terms:

(1/2) * [-cos(7*11x)/(7*11)] - (1/2) * [-cos(x)/1] + C

Simplifying the expression gives us the final result.

Formula Used

The integral of cos(4*11x) * sin(3*11x) dx is calculated using the following steps:

∫ cos(4*11x) * sin(3*11x) dx = (1/2) ∫ [sin(7*11x) - sin(x)] dx = (1/2) * [-cos(7*11x)/(7*11) + cos(x)/1] + C = [-cos(7*11x)/(154) + cos(x)/2] + C

Where C is the constant of integration.

Worked Example

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

∫[0 to π/2] cos(4*11x) * sin(3*11x) dx = [-cos(7*11*(π/2))/(154) + cos(π/2)/2] - [-cos(0)/(154) + cos(0)/2] = [-cos(377π/2)/154 + 0] - [-1/154 + 1/2] = -cos(377π/2)/154 + 1/154 - 1/2 = (-cos(377π/2) - 77)/154

Since cos(377π/2) = cos(188π + π/2) = cos(π/2) = 0, the result simplifies to -77/154.

Interpreting the Result

The result of the integral represents the area under the curve of cos(4*11x) * sin(3*11x) between the specified limits. For definite integrals, the result is a single numerical value representing the net area. For indefinite integrals, the result includes the constant of integration C.

Frequently Asked Questions

What is the integral of cos(411x) sin(3*11x) dx?

The integral is calculated using trigonometric identities to simplify the product of cosine and sine functions before integration. The result is expressed in terms of cosine functions.

How do I compute the definite integral of this function?

To compute the definite integral, apply the antiderivative to the upper and lower limits, then subtract the results. The calculator handles this automatically when you input the limits.

What are the assumptions made in this calculation?

The calculation assumes that the function cos(4*11x) * sin(3*11x) is continuous over the interval of integration. The result is valid for all real numbers x.

Can I use this calculator for complex numbers?

This calculator is designed for real-valued functions. For complex numbers, additional mathematical considerations are required.

How accurate is this calculator?

The calculator uses precise mathematical formulas and JavaScript's built-in trigonometric functions to ensure accurate results.