If 2 \\int(x^{2})+3 \\int({1}/{x^{2}})=x^{2}-1, \\forall x \R-\{0\} then \\int(x^{4}) is equal to

Correct Answer is : (1−x4)(2x4+3)5x4

CMAT 2021 Solved Question Paper

Question : 25 of 100

Marks: +1, -0

2 \∫(x^{2})+3 \∫({1}/{x^{2}})=x^{2}-1, \∀ x \R-\{0\}

then

\∫(x^{4})

is equal to

Solution:

Given,

2 \∫(x^{2})+3 \∫({1}/{x^{2}})=x^{2}-1

...(i)
Replacing

x

{1}/{x}

we get

2 \∫({1}/{x^{2}})+3 \∫(x)^{2}={1}/{x^{2}}-1

...(ii)
On multiplying Eq. (i) by 2, Eq. (ii) by 3 and then subtracting Eq. (i) from Eq. (ii), we get

5 \∫(x^{2})={3}/{x^{2}}-1-2 x^{2}

⇒

\∫(x^{2})={1}/{5 x^{2}}(3-x^{2}-2 x^{4})

∴

\∫(x^{4})={1}/{5 x^{4}}(3-x^{4}-2 x^{8})

(\replacing x \by x^2)

={{(1-x^{4})(2 x^{4}+3)}}/{5 x^{4}}

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