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)