SOLUTION: Please help me solve this equation: {{{ int( (1-x)^2010, dx, 0, 1 ) }}}

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Question 344815: Please help me solve this equation: +int%28+%281-x%29%5E2010%2C+dx%2C+0%2C+1+%29+
Found 2 solutions by Alan3354, Edwin McCravy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
= -%281-x%29%5E2011%2F2011+%2B+C
--> 0 + 1/2011
= 1/2011

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
I will put x=%22%22 on the limits to show what variable the
limits are on, namely x, since we will be changing the limits:

+int%28+%281-x%29%5E2010%2C+dx%2C+x=0%2C+x=1+%29+

Let u=1-x
Then %28du%29%2F%28dx%29=-1
or du=-dx

and dx=-du

So we substitute u for %281-x%29 and -du for dx

+int%28+u%5E2010%2C+%28-du%29%2C+x=0%2C+x=1+%29+

And we can take out the negative sign in front of the integral:

+-int%28+u%5E2010%2C+du%2C+x=0%2C+x=1+%29+

But the limits are on x.  We want to change the limits
so that they will be on u rather than x.

To do that we substitute the limits for x in the equation
for u which is u=1-x

when x=0, u=1-0=1 and
when x=1, u=1-1=0,
so we change the limits to:

+-int%28+u%5E2010%2C+du%2C+u=1%2C+u=0+%29+

and we use the power formula int%28+u%5En%2C+du%29=u%5E%28n%2B1%29%2F%28n%2B1%29%2BC
 


Edwin