If f : R R, be a continuous function such that f(x) = _1^x tf(t)d… - Vidyadip

MCQ

If $f : R \to R,$ be a continuous function such that $f(x) = \int\limits_1^x {tf(t)dt,}$ then the correct statement is -

A
$\int\limits_{ - \pi }^x {f(x)dx = 2\pi }$
B
$\int\limits_{ - \pi }^x {f(x)dx = \pi }$
✓
$\int\limits_{ - 3}^3 {f(x)dx = 0}$
D
$\int\limits_{ - 3}^3 {f(x)dx = 12}$

✓

Answer

Correct option: C.

$\int\limits_{ - 3}^3 {f(x)dx = 0}$

c
$f^{\prime}(x)=f(x) ; f(1)=0$

$\therefore \frac{f^{\prime}(\mathrm{x})}{f(\mathrm{x})}=1 \Rightarrow \ln (f(\mathrm{x}))=\mathrm{x}+\mathrm{c}$

$\Rightarrow f(\mathrm{x})=\mathrm{k} \cdot \mathrm{e}^{\mathrm{x}} \Rightarrow \mathrm{k}=0$

$\therefore f(\mathrm{x})=0$

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