anonymous 5 years ago How do I solve the integral of x(x-10)^10?

1. anonymous

do a substitution with u=10-x

2. anonymous

sorry kill that u=x-10 that should say

3. anonymous

du/dx = 1

4. anonymous

wait

5. anonymous

that wont work

6. anonymous

this isnt hard

7. anonymous

use binomial formula to expand (x-10)^10

8. anonymous

no i see the problem, yeah it seemed to easy

9. anonymous

integral x [ x^10 + 10 choose 1 x^9 (-1) + 10 choose 2 x^8 * (-1)^2 + ...

10. anonymous

make a pascal triangle

11. anonymous
12. anonymous

online whiteboard

13. anonymous

i think you should be able to use the substitution u=x-10 du/dx = 1 so this is the integral of (u+10)u^10 which is simply u^11 +10u^10 so this gives soln (u^12)/12 +10(u^11)/11 and then put x back in for u

14. anonymous

oh

15. anonymous

You have to use a complex substitution: $\int\limits_{}^{} x(x-10)^{10} dx$ u = x-10 x = u + 10, dx = 1 du Substitute and profit!

16. anonymous

17. anonymous

it doesn't have to be complex this problem is purely in the reals

18. anonymous

ac, right. i think he meant . wrong choice of words

19. anonymous

Oh, I meant complex as in 'difficult', lol.

20. anonymous

hehe

21. anonymous

i became both of you guys fans

22. anonymous

acland, it would be much tougher if you had to do integral x^10 (x-10)^10 dx

23. anonymous

then there is no avoiding pascal's beautiful triangle

24. anonymous

bad choice of words, you get to used to dealing with complex number systems all the time. For the x^2 one you should be able to use the same idea and have the integral of $(u+10)^2+u^{10}$ yeah if it was x^10 then you would have to look at other ways of solving it

25. anonymous

u = x-10 du = dx u + 10 = x so int (u+10) u^10

26. anonymous

if it was x^10 i just wouldnt bother and would leave my final answer as an integral

27. anonymous

so int u^11 + 10 u^10

28. anonymous

i got a toughie

29. anonymous

A tract of land bordered by a highway along the y-axis, a dirt road along the x-axis, and a river whose path is given by the equation y=4-0.2x^2, where x and y are in hindreds of meters. The tract is 300m deep along the dirt road The value of the land is constant in any strip parallel to the highway and increases as you move way from the highway, with the value given by v=1000+50x dollars per 10,000 m^2 at the sample point (x,y). Find dW, the worth of a strip. Write an integral that equals the worth of the entire tract.

30. anonymous

i thought about using a double integral