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

$9$

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B

$3$

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C

$\frac{1}{9}$

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D

$\frac{1}{3}$

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Solution

## The correct option is A $9$Explanation for the correct option:Integration by substitution:Let $\begin{array}{rcl}3x& =& t\end{array}$, then $\begin{array}{rcl}dx& =& \frac{dt}{3}\end{array}$.$k{\int }_{0}^{3}\frac{t}{3}·f\left(t\right)\frac{dt}{3}={\int }_{0}^{3}t·f\left(t\right)dt\phantom{\rule{0ex}{0ex}}⇒\frac{k}{9}{\int }_{0}^{3}t·f\left(t\right)dt={\int }_{0}^{3}t·f\left(t\right)dt\phantom{\rule{0ex}{0ex}}⇒\frac{k}{9}=1\phantom{\rule{0ex}{0ex}}⇒k=9$Hence, option A is correct

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