Let f(x)=int x 3 sqrt 3- x 2 dx . If 5 f(√(2))=-4 , then f(1) is equal to

Mathematics · Indefinite Integration

JEE Main 2025 — 3 April, Morning Shift — Question 34

Let $f(x)=\int \mathrm{x}^{3} \sqrt{3-\mathrm{x}^{2}} \mathrm{dx}$ . If $5 f(\sqrt{2})=-4$ , then $f(1)$ is equal to

Let $3-x^{2}=t^{2}$

$+\mathrm{xdx}=-\mathrm{tdt}$

$f(x)=\int\left(3-t^{2}\right) \cdot t(-t d t)+c$

$=\int\left(\mathrm{t}^{4}-3 \mathrm{t}^{2}\right) \mathrm{dt}+\mathrm{c}$

$=\frac{\mathrm{t}^{5}}{5}-\mathrm{t}^{3}+\mathrm{c}$

$f(x)=\frac{\left(3-x^{2}\right)^{5 / 2}}{5}-\left(3-x^{2}\right)^{3 / 2}+c$

$f(\sqrt{2})=\frac{1}{5}-1+c=-\frac{4}{5}$

$\mathrm{c}=0$

$f(1)=\frac{2^{5 / 2}}{5}-2^{3 / 2}$

$=2^{1 / 2}\left(\frac{4}{5}-2\right)$

$f(1)-\frac{-6 \sqrt{2}}{5}$

Answer key and solution verified before publishing.

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