∫011-x9dx if it exists
By using the substitution rule for definite integrals and fundamental rule of calculus
2) Theorem and Rule:
Fundamental theorem of calculus:
If f is continuous on [a, b], then ∫abfxdx=Fb-F(a).
Substitution rule for definite integrals:
If g' is continuous on [a, b] and f is continuous on the range of u=g(x) then
Since the function is continuous in the given interval the integral exists.
By applying the substitution rule for definite integrals
Let u=1-x. Then du=-dx, so dx=-du
When x=0, u=1 and x=1, u=0
By using the fundamental theorem of calculus and power rule,