How do I solve this function?

Question

2
ò (x - 1)^9 dx =
0

Answer 1

lets do the question in a diff style grab ur pen and paper ....


plot a graph for (x-1)^3 since the function is odd

(x-1)^9 is also odd so the function is symetric about x=1 ...i.e it has a part of it in positive y quadrants and some in negative y quadrants...

since the function is placed over equal length the ( 1 unit on both sides) the function's integral becomes zero since positive integral cancels negative...this is graphical approach really useful some times use it often for simple fnctions

Answer 2

I think you mean the integral from 0 to 2 of (x-1)^9. Well, you must use u substitution. u=x-1, so du=dx. Thus the integrand becomes u^9. The new endpoints of integration are -1 and 1 (u=x-1). So integrate u^9 from -1 to 1. This would be (u^10)/10 from -1 to 1. This = 1/10 -1/10 =0.

Answer 3

now if it is the definite intergral from 0 to 2, we have to make a substitution x-1 = t
so, dx=dt
and the limits of t are from -1 to 1
so, the integral is from -1 to 1 u^9du
since u^9 is an odd function, the definite integral from -a to a will be 0
so, the answer is 0