Integrating x * (cos x)^ n between pi/2 and 0

Thread starter ZakS
Start date Aug 5, 2011
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Integrating

In summary, the conversation discusses the meaning, purpose, calculation, significance, and real-world applications of integrating the function x * (cos x)^n between the limits of pi/2 and 0. This involves finding the definite integral of the function within the specified interval, and can be useful in various fields such as physics, engineering, economics, and finance.

Aug 5, 2011

ZakS

Would anyone be able to help me do this? I have tried by parts, but did not make progress. As n gets large, the area gets smaller.

Your help is appreciated.

Z

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Aug 5, 2011

ehild

Homework Helper

You might apply that cos(x)=(e^ix+e^-ix)/2

ehild

Aug 6, 2011

Bohrok

Try integration by parts just once and use the formula for ∫cosⁿx dx twice.

Related to Integrating x * (cos x)^ n between pi/2 and 0

1. What is the meaning of "Integrating x * (cos x)^n between pi/2 and 0"?

The phrase "integrating x * (cos x)^n between pi/2 and 0" refers to finding the definite integral of the function x * (cos x)^n from the upper limit of pi/2 to the lower limit of 0. This involves calculating the area under the curve of the function within the specified interval.

2. What is the purpose of integrating x * (cos x)^n between pi/2 and 0?

The purpose of integrating x * (cos x)^n between pi/2 and 0 is to evaluate the total value of the function within the given interval. This can be useful in various applications, such as calculating work done or determining average values.

3. How is the definite integral of x * (cos x)^n between pi/2 and 0 calculated?

The definite integral of x * (cos x)^n between pi/2 and 0 can be calculated using integration techniques such as substitution, integration by parts, or trigonometric identities. The specific method used will depend on the complexity of the function and the desired level of accuracy.

4. What is the significance of the limits pi/2 and 0 in the integral of x * (cos x)^n?

The limits pi/2 and 0 represent the upper and lower bounds of the interval over which the function is being integrated. In this case, the integration is being performed over the interval where cos x is positive, which is from pi/2 to 0. This choice of limits ensures that the function is continuous and well-defined within the interval.

5. What are some real-world applications of integrating x * (cos x)^n between pi/2 and 0?

The integral of x * (cos x)^n between pi/2 and 0 can be applied in fields such as physics and engineering to calculate the work done by a varying force or the average value of a periodic function. It can also be used in economics and finance to determine the net present value of investments with changing returns over time.