Question

If $k \int_{0}^{1} x f(3x) dx = \int_{0}^{3} t f(t) dt$ then $k =$

• A. 9
• B. 3
• C. 1/9
• D. 1/3

Hint:

When replacing $x$ with $ax$ in the argument, the factor outside the integral often involves $a^2$ if there is an additional $x$ term.

Answer 1

The Correct Option is A

Step 1: Concept
Use the method of substitution for definite integrals to transform the variable of integration.

Step 2: Meaning
Let $t = 3x$. Then $dt = 3dx$, which means $dx = dt/3$. Also, $x = t/3$.

Step 3: Analysis
Change the limits: when $x=0$, $t=0$; when $x=1$, $t=3$. The integral becomes $k \int_{0}^{3} (t/3) f(t) (dt/3) = (k/9) \int_{0}^{3} t f(t) dt$.

Step 4: Conclusion
Comparing $(k/9) \int_{0}^{3} t f(t) dt$ with $\int_{0}^{3} t f(t) dt$, we get $k/9 = 1$, which implies $k = 9$. Final Answer: (A)