JEE Mains · Maths · STD 12 - 7.2 definite integral
Let for some function \(\mathrm{y}=f(x), \int_0^x t f(t) d t=x^2 f(x), x\gt0\) and \(f(2)=3\). Then \(f(6)\) is equal to
- A 1
- B 3
- C 6
- D 2
Answer & Solution
Correct Answer
(A) 1
Step-by-step Solution
Detailed explanation
\begin{aligned} & \int_0^x t f(t) d t=x^2 f(x) \\ & \Rightarrow \quad x f(x)=2 x f(x)+x^2 f^{\prime}(x) \\ & \Rightarrow x^2 \cdot f^{\prime}(x)=-x f(x) \\ & \Rightarrow \quad \frac{f^{\prime}(x)}{f(x)}=-\frac{1}{x} \\ & \\ & \quad \int \frac{f^{\prime}(x)}{f(x) d x}=-\int…
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