Let f(x)=int 0 x t (t 2 -9 t+20 ) d t, 1 leq x leq 5 . If the range of f is [α, β] , then 4(α+β) equals:

Mathematics · Definite Integration

JEE Main 2025 — 29 January, Evening Shift — Question 59

Let $f(x)=\int_{0}^{x} t\left(t^{2}-9 t+20\right) d t, \quad 1 \leq x \leq 5$ . If the range of $f$ is $[\alpha, \beta]$ , then $4(\alpha+\beta)$ equals:

Option A:
157
Correct
Option B:
253
Option C:
125
Option D:
154

Answer: A

Step-by-step solution

$f^{\prime}(x)=x^{3}-9 x^{2}+20 x=x(x-4)(x-5)$

$\therefore \mathbf{f}(\mathbf{x})=\frac{\mathrm{x}^{4}}{4}-\frac{9 \mathrm{x}^{3}}{3}+\frac{20 \mathrm{x}^{2}}{2}$

$f(1)=\frac{1}{4}-3+10=\frac{29}{4}=\alpha$

$\left.f(4)=\frac{256}{4}-3(64) \right\rvert\,+10(16)=32=\beta$

$4(\alpha+\beta)=4\left(\frac{29}{4}+32\right)=157$

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Exam: JEE Main 2025
Paper: 29 January, Evening Shift
Subject: Mathematics
Chapter: Definite Integration
Topic: Introduction to Definite Integration

If 24 int 0^pi/4 (sin 4 x-pi/12 +[2 sin x] ) d x=2 pi+alpha , where [.] denotes the greatest integer function, then alpha is equal to

Answer: A

Step-by-step solution

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