Question

Find the validity of the statement: $\int\limits_{a}^{b}{f\left( x \right)dx}=-\int\limits_{a+c}^{b+c}{f\left( x-c \right)dx}$.
A. true
B. false

Answer 1

Hint: We need to change the variable of the given relation. We assume the relation $z=x+c$. We then change the limits according to the relation. We use the theorem of definite integral that $\int\limits_{a}^{b}{f\left( x \right)dx}=\int\limits_{a}^{b}{f\left( z \right)dz}$ to find the required relation. We validate it with the given one and find the correct option.

Complete step-by-step solution
We have the theorem of definite integral that $\int\limits_{a}^{b}{f\left( x \right)dx}=\int\limits_{a}^{b}{f\left( z \right)dz}$.
Now we have an equation of integral where we have to prove $\int\limits_{a}^{b}{f\left( x \right)dx}=-\int\limits_{a+c}^{b+c}{f\left( x-c \right)dx}$.
We try to form a function where we take another variable z dependent on x.
The relation is $z=x+c$.
Solving the relation, we find value of x as
$\begin{align}
  & z=x+c \\
 & \Rightarrow x=z-c \\
\end{align}$
We take differential both side to find, c being constant
$\begin{align}
  & z=x+c \\
 & \Rightarrow dz=dx \\
\end{align}$
Now we try to find the limits of the new variable.

x	a	b
\[z=x+c\]	\[a+c\]	\[b+c\]

So, the limits change to \[a+c\] and \[b+c\].
We replace the variable on the left side of the equation.
$\int\limits_{a}^{b}{f\left( x \right)dx}=\int\limits_{a+c}^{b+c}{f\left( z-c \right)dz}$.
Here we just changed the variable values from their respective relations.
Now we have the theorem of $\int\limits_{a}^{b}{f\left( x \right)dx}=\int\limits_{a}^{b}{f\left( z \right)dz}$.
We apply the theorem on the above equation to get
$\begin{align}
  & \int\limits_{a}^{b}{f\left( x \right)dx}=\int\limits_{a+c}^{b+c}{f\left( z-c \right)dz} \\
 & \Rightarrow \int\limits_{a}^{b}{f\left( x \right)dx}=\int\limits_{a+c}^{b+c}{f\left( x-c \right)dx} \\
\end{align}$
We just change the variable form of the integration.
Now we try to validate the statement. We can identify that there is an extra negative sign in the given relation. So, the given statement is wrong. The correct option is B.

Note: We need to understand that we are changing the variables twice using two different methods. The first time we are using a relation or function $z=x+c$. The value of the integration remains the same as the limits also change with the variable change. But the second time we used the theorem of definite integral to form the statement.