Posted by **Ethan** on Wednesday, September 28, 2011 at 12:00am.

given the graph of f(x) = x sinx, 0<=x<=2pi

assuming that a quantity y changes at a rate of y' = xsinx, find by how much it will increase or decrease over 3pi/2 <= x <= 2pi

- calculus -
**Steve**, Wednesday, September 28, 2011 at 10:46am
This is just a problem in integrals. We have y', and we want to add up all the small changes given by that function.

Int[x sinx dx] can be solved using integration by parts.

Recall the product rule of derivatives:

(uv)' = u'v + uv'

so

uv' = (uv)' - u'v

Going the other direction,

Int[u dv] = uv - Int[v du]

So, we want to split up the integrand into two factors, where one part (u) gets simpler after differentiation, and the other part (dv) can be easily integrated.

Here, we have x sinx

If u = x, then du = dx

If dv = sinx, then v = -cosx

Int[x sinx dx] = uv - Int[v du]

= -x cosx - Int[-cos x * dx]

Now we have a simple integrand, -cosx

Int cos x dx = sin x

So, the final integration is

Int[x sinx dx] = uv - Int[v du]

= -x cosx - Int[-cos x * dx]

= -x cosx + sin x

Evaluating at 3pi/2 and 2pi, we have

(-2pi * 1 + 0) - (-3pi/2 * 0 + -1) = -2pi + 3pi/2 + 1 = 1 + 7pi/2

## Answer This Question

## Related Questions

- Please Help!!! - given the graph of f(x) = x sinx, 0<=x<=2pi assuming ...
- calculus - given the graph of f(x) = x sinx, 0<=x<=2pi assuming that a ...
- trig - Solve cos x-1 = sin^2 x Find all solutions on the interval [0,2pi) a. x=...
- CALC HW - FIND THE X VALUES OF ALL POINTS, -2PI LESS THAN OR EQUAL TO X LESS ...
- Calculus - Find all points of extrema on the interval [0, 2pi] if y=x+sinx (-1.-...
- Pre-Calculus-check answers - State the period and phase shift of the function y...
- math - Hello there, im doing an equation and its mind boggling me badly so i ...
- Math - This is on the 2008 Free response AP Calc AB exam- #4a. Let f be the ...
- Trigonometry - Graph the function y=2sin(x-2pi/3). To draw the graph, plot all x...
- Trigonometry - I really need help with this trigonometry question.please any ...

More Related Questions