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STD 12 - 7.2 definite integral — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 7.2 definite integral4 Marks
MCQ
If $x \phi(x)=\int_{5}^{x}\left(3 t^{2}-2 \phi^{\prime}(t)\right) d t, x\,>\,-2$, and $\phi(0)=4$ then $\phi(2)$ is .... .
  • ✓
    $4$
  • B
    $6$
  • C
    $8$
  • D
    $10$
✓

Answer

Correct option: A.
$4$
a
$\mathrm{x} \phi(\mathrm{x})=\int_{5}^{\mathrm{x}} 3 \mathrm{t}^{2}-2 \phi^{\prime}(\mathrm{t}) \mathrm{dt}$

$\mathrm{x} \phi(\mathrm{x})=\mathrm{x}^{3}-125-2[\phi(\mathrm{x})-\phi(5)]$

$\mathrm{x} \phi(\mathrm{x})=\mathrm{x}^{3}-125-2 \phi(\mathrm{x})-2 \phi(5)$

$\phi(0)=4 \Rightarrow \phi(5)=-\frac{133}{2}$

$\phi(\mathrm{x})=\frac{\mathrm{x}^{3}+8}{\mathrm{x}+2}$

$\phi(2)=4$

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