Evaluate d∫1+x1-xdx, x ≠1

Question

Evaluate `int sqrt((1 + x)/(1 - x)) "d"x`, x ≠1

Sum

Solution

Let I = `int sqrt((1 + x)/(1 - x)) "d"x`

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

= `sin^-1x + 1`

When I₁ = `(x"d"x)/sqrt(1 - x^2)`.

Put 1 – x² = t²

⇒ –2x dx = 2t dt.

Therefore I₁ = – dt = – t + C

= `- sqrt(1 - x^2) + "C"`

Hence I = `sin^-1x - sqrt(1 - x^2) + "C"`.

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Definite Integrals

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Chapter 7: Integrals - Solved Examples [Page 147]

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Chapter 7 Integrals
Solved Examples | Q 4 | Page 147

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Evaluate d∫1+x1-xdx, x ≠1

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