KCET · Maths · Definite Integration
If \(k \int_{0}^{1} x \cdot f(3 x) d x=\int_{0}^{3} t \cdot f(t) d t\), then the value of \(k\) is
- A 9
- B 3
- C \(\frac{1}{9}\)
- D \(\frac{1}{3}\)
Answer & Solution
Correct Answer
(A) 9
Step-by-step Solution
Detailed explanation
Given,
when \(x=1, t=3\)
\(\therefore \quad I=k \int_{0}^{3} \frac{t}{3} \cdot f(t) \cdot \frac{d t}{3}\)
Now, form Eq. (i), we get
\[
\begin{aligned}
\frac{k}{9} \int_{0}^{3} t \cdot f(t) d t &=\int_{0}^{3} t \cdot f(t) \\
\frac{k}{9}=1 \Rightarrow k=9
\end{aligned}
\]
when \(x=1, t=3\)
\(\therefore \quad I=k \int_{0}^{3} \frac{t}{3} \cdot f(t) \cdot \frac{d t}{3}\)
Now, form Eq. (i), we get
\[
\begin{aligned}
\frac{k}{9} \int_{0}^{3} t \cdot f(t) d t &=\int_{0}^{3} t \cdot f(t) \\
\frac{k}{9}=1 \Rightarrow k=9
\end{aligned}
\]
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