which of the following is equivalent to integral (a,b) k*f(x)+C)dx where k and C are constants
k integral (a,b)(f(x)+C)dx *****
intergral (a,b)kdx + intergral (a,b)f(x)dx+ intergral (a,b) Cdx
k integral (a,b)f(x)+ integral (a,b) Cdx
integral (a,b) kdx * integral (a,b) f(x) dx +integral (a,b) Cdx
nope, you need k * f(x) and C alone.
∫k*f(x)+C dx
= ∫k*f(x) dx + ∫C dx
= k∫f(x) dx + ∫C dx
your answer is ∫k(f(x)+C) dx
To determine which of the options is equivalent to the given integral ∫(a,b) k*f(x) + C dx, let's break down the steps.
First, let's recall the linearity property of integration and the constant multiple rule:
∫(a,b) (kf(x) + C) dx = k∫(a,b) f(x) dx + ∫(a,b) C dx
Using this property, we can simplify the options:
Option 1: k∫(a,b) (f(x) + C) dx
Option 2: ∫(a,b) k dx + ∫(a,b) f(x) dx + ∫(a,b) C dx
Option 3: k∫(a,b) f(x) dx + ∫(a,b) C dx
Option 4: ∫(a,b) k dx * ∫(a,b) f(x) dx + ∫(a,b) C dx
Now let's compare each option to the original integral:
Option 1 is equivalent to the original integral, as it correctly applies the linearity property.
Option 2 is not equivalent because the integral of a constant C with respect to x is simply Cx, not ∫(a,b) C dx.
Option 3 is equivalent to the original integral, as it correctly applies the linearity property.
Option 4 is not equivalent because multiplying two integrals together is not a valid operation in this context.
So the correct answer is Option 1: k∫(a,b) (f(x) + C) dx.