MHT CET 2023 13th May Morning Shift | Indefinite Integration Question 16 | Mathematics

If $$\int x^5 e^{-4 x^3} \mathrm{~d} x=\frac{1}{48} \mathrm{e}^{-4 x^3} \mathrm{f}(x)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration, then $$\mathrm{f}(x)$$ is given by

$$4 x^3+1$$

$$-4 x^3-1$$

$$-2 x^3-1$$

$$-2 x^3+1$$

MHT CET 2023 13th May Morning Shift

MCQ (Single Correct Answer)

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If $$\mathrm{f}(x)=\int \frac{x^2 \mathrm{~d} x}{\left(1+x^2\right)\left(1+\sqrt{1+x^2}\right)}$$ and $$\mathrm{f}(0)=0$$, then $$\mathrm{f}(1)$$ is

$$\log (1+\sqrt{2})$$

$$\log (1+\sqrt{2})-\frac{\pi}{4}$$

$$\log (1+\sqrt{2})+\frac{\pi}{4}$$

$$\log (1-\sqrt{2})$$

MHT CET 2023 12th May Evening Shift

MCQ (Single Correct Answer)

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$$\int \frac{1}{\cos ^3 x \sqrt{\sin 2 x}} d x=$$

$$\sqrt{2}\left(\sqrt{\tan x}+\frac{1}{5}(\tan x)^{\frac{5}{2}}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.

$$\left(\sqrt{\tan x}+\frac{2}{5}(\tan x)^{\frac{5}{2}}\right)+c$$, where c is a constant of integration.

$$\frac{1}{\sqrt{2}}\left(\sqrt{\tan x}+\frac{2}{5}(\tan x)^{\frac{5}{2}}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.

$$2\left(\sqrt{\tan x}+\frac{1}{5}(\tan x)^{\frac{5}{2}}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.

MHT CET 2023 12th May Evening Shift

MCQ (Single Correct Answer)

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If $$\int \frac{\sqrt{1-x^2}}{x^4} \mathrm{~d} x=\mathrm{A}(x)\left(\sqrt{1-x^2}\right)^{\mathrm{m}}+\mathrm{c}$$ for a suitable chosen integer $$\mathrm{m}$$ and a function $$\mathrm{A}(x)$$, where $$\mathrm{c}$$ is a constant of integration, then $$(\mathrm{A}(x))^{\mathrm{m}}$$ equals

$$\frac{1}{9 x^4}$$

$$\frac{-1}{3 x^3}$$

$$\frac{-1}{27 x^9}$$

$$\frac{1}{27 x^6}$$

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