Question

If $k{\int }_0^{1}x·f\left(3x\right)dx={\int }_0^{3}t·f\left(t\right)dt$, then the value of $k$ is:

• A. $9$
• B. $3$
• C. $\frac{1}{9}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A

$9$

Explanation for the correct option:

Integration by substitution:

Let $\begin{array}{rcl}3x& =& t\end{array}$, then $\begin{array}{rcl}dx& =& \frac{dt}{3}\end{array}$.

$$\begin{gathered} k{\int }_0^3\frac{t}{3}·f\left(t\right)\frac{dt}{3}={\int }_0^{3}t·f\left(t\right)dt \\ \Rightarrow \frac{k}{9}{\int }_0^{3}t·f\left(t\right)dt={\int }_0^{3}t·f\left(t\right)dt \\ \Rightarrow \frac{k}{9}=1 \\ \Rightarrow k=9 \end{gathered}$$

Hence, option A is correct