Show that if f(x) is the antiderivative of v(x) then f(2x) is the...

Question:

Show that if {eq}f(x) {/eq} is the antiderivative of {eq}v(x) {/eq} then {eq}f(2x) {/eq} is the antiderivative of {eq}v(2x) {/eq}.

Anti-derivatives:

Anti-derivatives, which are necessary to perform calculations for indefinite integrals or definite integrals, apply operations that the opposite effect on an expression than applying the derivative on the expression would.

Answer and Explanation:

Suppose if {eq}v(x) = 0 {/eq}, {eq}f(x) {/eq} can be found by integrating {eq}v(x) {/eq}


{eq}\begin{align*} f(x) &= \int v(x) dx \\ &= \int 0 dx \\ &= 0+c \text{ [Constant of integration added due to indefinite integral present]} \\ &= c \\ \end{align*} {/eq}


When {eq}v(2x) {/eq}:

{eq}v(x) = 0 \Rightarrow v(2x) = 0 {/eq}.


When {eq}f(2x) {/eq}:

{eq}f(x) = c \Rightarrow f(2x) = c {/eq}


{eq}v(2x) {/eq} will be integrated to confirm that {eq}f(2x) {/eq} is the anti-derivative of {eq}v(2x) {/eq}.


{eq}\begin{align*} f(2x) &= \int v(2x) dx \\ &= \int 0 dx \\ &= 0+c \text{ [Constant of integration added due to indefinite integral present]} \\ &= c \text{ [Same as when 2x was the input of f(x)]} \\ \end{align*} {/eq}


This shows that the statement is satisfied when {eq}v(x) = 0 {/eq}. Polynomials, natural logarithms, and trigonometric components in the expression will not lead to both {eq}f(x) {/eq} being the antiderivative of {eq}v(x) {/eq} and {eq}f(2x) {/eq} and {eq}v(2x) {/eq} due to the variable substitution and chain rule for integrals being applied when calculating the antiderivative.


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Antiderivative: Rules, Formula & Examples

from Calculus: Help and Review

Chapter 8 / Lesson 12
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