How Can You Integrate xsinxcosxdx Using Exponential Form?

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Discussion Overview

The discussion revolves around the integration of the function xsin(x)cos(x)dx, exploring various methods including integration by parts and the use of exponential forms of trigonometric functions. The scope includes mathematical reasoning and techniques for integration.

Discussion Character

  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant suggests using integration by parts with u=x and dv=sin(x)cos(x)dx, leading to a formulation involving (sin(x))^2.
  • Another participant proposes rewriting the integrand using the identity 2sin(x)cos(x) = sin(2x), suggesting a substitution y=2x to facilitate integration.
  • A different approach is introduced, advocating for the use of exponential forms of sine and cosine to express the integrand, implying that this method could simplify the integration process.

Areas of Agreement / Disagreement

Participants present multiple competing views on how to approach the integration, with no consensus on a single method being preferred or more effective.

Contextual Notes

Some methods rely on specific identities and substitutions that may not be universally applicable without further context or conditions. The discussion does not resolve the effectiveness of each proposed method.

Who May Find This Useful

Mathematics students or enthusiasts interested in integration techniques, particularly those involving trigonometric functions and their exponential forms.

carlosgrahm
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How do you integrate

xsinxcosxdx
 
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Inegrate by parts: u=x, dv=sinxcosxdx=sinxd(sinx)
You get x(sinx)2/2 -integral of (1/2)(sinx)2dx

You should be able to proceed (using double angle formula for cos to get rid of (sinx)2/2).
 
Since
2sin(x)cos(x) = sin(2x)
you can write the integrand function
x/2 \cdot \sin (2x)
you can use first the substitution
y=2x
and then use integration by part formula to integrate
y/4 \cdot \sin (y)
it is EASY if you choose to derive y/4 and integrate \sin(y).
 
You can solve any question like this by expressing sin(x), cos(x), etc in terms of their exponential form and multiplying everything out.

cos(x) = [exp(ix)+exp(-ix)]/2
sin(x) = [exp(ix)-exp(-ix)]/(2i)
 

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