What is the solution to the integral of xSin(x)Sin(2x) dx?

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In summary, After some thought and calculations, the solution to the integral ∫x*Sin(x)*Sin(2x) dx can be found by using the IBP formula and the trig identity 2 sin x cos x = sin 2x. The final answer is Sin(x) - xSin(2x)Cos(x) + C.
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PrudensOptimus
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A few weeks ago, I saw a post regarding the area under the function xsinxsin2x dx, (x = pi*x/a). After cogitations, I have found the answer:

∫x*Sin(x)*Sin(2x) dx = - xSin(2x)Cos(x) + ∫Cos(x) dx
= Sin(x) - xSin(2x)Cos(x) + C

By assuming u = xsin2x, du = 2xcos2x + sin2x, dv = sinx, v = -cosx. And using the Parts formula.
 
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  • #2
The IBP formula is

∫ u dv = uv - ∫ v du

not ∫ v dx


BTW, the key to this is the trig identity
2 sin x cos x = sin 2x
 
  • #3


The solution to the integral of xSin(x)Sin(2x) dx is Sin(x) - xSin(2x)Cos(x) + C. This was found by using the integration by parts formula and assuming u = xsin2x and dv = sinx. After solving for du and v, the formula was used to find the solution. It is important to note that the constant C was added to account for the indefinite integral. This solution can be verified by taking the derivative, which will result in the original function.
 

1. What is the meaning of the integral ∫ xSin(x)Sin(2x) dx?

The integral ∫ xSin(x)Sin(2x) dx represents the area under the curve of the function xSin(x)Sin(2x) with respect to the variable x. It is a mathematical calculation used in calculus to find the exact value of a function over a given interval.

2. How do you solve the integral ∫ xSin(x)Sin(2x) dx?

To solve this integral, we use integration techniques such as integration by parts or substitution. First, we simplify the integrand by using the trigonometric identity Sin(2x) = 2Sin(x)Cos(x), which gives us ∫ 2xSin(x)Cos(x)Sin(x) dx. Then, we use integration by parts with u = x and dv = Sin(x)Cos(x)dx to find the antiderivative of the integrand. Finally, we evaluate the integral over the given interval.

3. What is the domain of the function xSin(x)Sin(2x)?

The domain of the function xSin(x)Sin(2x) is all real numbers.

4. What is the range of the function xSin(x)Sin(2x)?

The range of the function xSin(x)Sin(2x) is the set of all real numbers between -1 and 1, inclusive.

5. What is the significance of the integral ∫ xSin(x)Sin(2x) dx in real-world applications?

The integral ∫ xSin(x)Sin(2x) dx has various applications in physics, engineering, and economics. For example, in physics, it can be used to calculate the work done by a force on an object moving in a circular path. In economics, it can be used to determine the profit or loss of a company over a given period of time. In general, this integral helps to find the total accumulation of a function over a given interval, which has many practical applications.

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