How Can You Integrate xsinxcosxdx Using Exponential Form?

[carlosgrahm]

How do you integrate

xsinxcosxdx

 

[mathman]

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).

 

[Take_it_Easy]

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).

 

[maze]

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)