Let , where C is the constant of integration. If , then equals: [2025]

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Solution

Q.
Let $\int x^{3} \sin x d x = g (x) + C$ , where C is the constant of integration. If $8 (g (\frac{π}{2}) + g' (\frac{π}{2})) = α π^{3} + {βπ}^{2} + γ, α, β, γ \in Z$ , then $α + β - γ$ equals: [2025]

1 47

2 62

3 48

4 55

Ans.
(4)

$\int x^{3} \sin x d x = - x^{3} \cos x + \int 3 x^{2} \cos x d x$

$= - x^{3} \cos x + 3 x^{2} \sin x - \int 6 x^{} \sin x d x$

$= - x^{3} \cos x + 3 x^{2} \sin x + 6 x \cos x - 6 \sin x + C$

So, $g (x) = - x^{3} \cos x + 3 x^{2} \sin x + 6 x \cos x - 6 \sin x$

and $g' (x) = - 3 x^{2} \cos x + x^{3} \sin x + 3 x^{2} \cos x + 6 x \sin x - 6 x \sin x + 6 \cos x - 6 \cos x$

$= x^{3} \sin x$

$\Rightarrow g (\frac{π}{2}) = \frac{3 π^{2}}{4} - 6 and g' (\frac{π}{2}) = \frac{π^{3}}{8}$

$∴ 8 (g (\frac{π}{2}) + g' (\frac{π}{2})) = π^{3} + 6 π^{2} - 48$

$\Rightarrow α = 1, β = 6, γ = - 48$

So, $α + β - γ = 55$ .

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