The given definite integral.
i) The substitution rule: for definite integral: If g'(x) is a continuous function on a,b whose f is continuous on range of u=g(x), then ∫abfgxg'(x)dx=∫g(a)g(b)f(u)du. Here g(x) is substituted as u and then differentiation g’(x)dx =du
The given integral is
Here using the substitution method
Differentiating with respect to x
The limits changes; the new limits of integration are\ calculated by substituting
For x=0, u=0-12=1, and for x=2, u=2-12=1
Therefore, the given integral becomes