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

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The discussion focuses on the integration of the function x * (cos x)^n from 0 to π/2. The user Z seeks assistance with this integration, noting that as n increases, the area under the curve diminishes. User ehild suggests applying integration by parts once and utilizing the formula for ∫cos(nx) dx twice to simplify the process. This approach is confirmed as effective for handling the integration of trigonometric functions raised to a power.

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

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

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