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 ____.

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Definite Integrals

x ϕ (x) = \int_{5}^{x} (3 t^{2} - 2 ϕ^{'} (t)) d t, x > - 2

, and

ϕ (0) = 4

, then

ϕ (2)

is ____.

Answer: 4

Differentiating both sides with respect to

x

gives

ϕ (x) + x ϕ^{'} (x) = 3 x^{2} - 2 ϕ^{'} (x)

, so

(x + 2) ϕ^{'} (x) + ϕ (x) = 3 x^{2}

. The left side is

\frac{d}{d x} [(x + 2) ϕ (x)]

, hence

(x + 2) ϕ (x) = x^{3} + C

. With

ϕ (0) = 4

2 \cdot 4 = C

, so

C = 8

and

(x + 2) ϕ (x) = x^{3} + 8

. Then

ϕ (2) = \frac{8 + 8}{4} = 4

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