Final answer:
To prove that ∫ᵇₐ c⋅f(x)dx = c∫ᵇₐ f(x)dx, we can use the limit definition of an integral. Let's assume that the integral ∫ᵇₐ f(x)dx exists, and express it as a limit using the sum notation. By multiplying f(x) by a constant c, we can factor out the constant from the sum, showing that ∫ᵇₐ c⋅f(x)dx = c∫ᵇₐ f(x)dx.
Step-by-step explanation:
To prove that ∫ᵇₐ c⋅f(x)dx = c∫ᵇₐ f(x)dx, we need to use the limit definition of an integral.
Let's assume that the integral ∫ᵇₐ f(x)dx exists. Now, we can express this integral using a limit:
∫ᵇₐ f(x)dx = limn→∞ Σi=1n f(xi*)∆xi,
where f(xi*) is some value of f(x) between xi-₁ and xi, and ∆xi is the width of the subinterval.
Now, if we multiply f(x) by a constant c, we have c⋅f(x), and the limit definition of the integral becomes:
∫ᵇₐ c⋅f(x)dx = limn→∞ Σi=1n (c⋅f(xi*)∆xi).
As we can see, the constant c can be factored out of the sum:
∫ᵇₐ c⋅f(x)dx = c⋅limn→∞ Σi=1n (f(xi*)∆xi).
This is the same as c times the original integral ∫ᵇₐ f(x)dx. Therefore, we have proved that ∫ᵇₐ c⋅f(x)dx = c∫ᵇₐ f(x)dx.