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Evaluate d∫x2dxx4+x2-2 - Mathematics

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Question

Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`

Sum

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Solution

Let x² = t.

Then `x^2/(x^4 + x^2 - 2) = "t"/("t"^2 + "t" - 2)`

= `"t"/(("t" + 2)("t" - 1))`

= `"A"/("t" + 2) + "B"/("t" - 1)`

So t = A(t – 1) + B(t + 2)

Comparing coefficients, we get A = `2/3`, B = `1/3`.

So `x^2/(x^4 + x^2 - 2) = 2/3 1/(x^2 + 2) + 1/3 1/(x^2 - 1)`

Therefore, `int x^2/(x^4 + x^2 - 2) "d"x`

= `2/3 int 1/(x^2 + 2) "d"x + 1/3 int "dx"/(x^2 - 1)`

= `2/3 1/sqrt(2) tan^-1 x/sqrt(2) + 1/6 log |(x + 1)/(x + 1)| + "C"`

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Definite Integrals

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Chapter 7: Integrals - Solved Examples [Page 154]

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NCERT Exemplar Mathematics [English] Class 12

Chapter 7 Integrals
Solved Examples | Q 15 | Page 154

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