Show integral(a,b)(f(x+c)dx = integral(a+c,b+c)(f(x) dx.
Let u = x+c, then du = dx so integral(a,b)(f(x+c)dx = integral(a+c,b+c)f(u)du. But when I substitute u back in, I get integral(a+c,b+c)(f(x+c)(du) which does not equal integral(a+c,b+c)(f(x)dx. What am I doing wrong?
As TPH said:
Let . Then the lower bound on u is and the upper bound on u is . Thus
as you desire.
Once you change the variable in the integrand, you need to do the same variable change in the limits of the integral to be consistent.
Now you can change the dummy variable u back to x:
but note again that the integrand has been rewritten so that the arguments of f are different.
-Dan