Question

The value of `int ((x^2 - 1)dx)/(x^3 sqrt(2x^4 - 2x^2 + 1)` is ______.

पर्याय

• `2sqrt(2 - 2/x^2 + 1/x^4) + C`

• `2sqrt(2 + 2/x^2 + 1/x^4) + C`

• `1/2sqrt(2 - 2/x^2 + 1/x^4) + C`

• None of these

Answer 1

The value of `int ((x^2 - 1)dx)/(x^3 sqrt(2x^4 - 2x^2 + 1)` is `underlinebb(1/2sqrt(2 - 2/x^2 + 1/x^4) + C)`.

Explanation:

Let I = `int ((x^2 - 1)dx)/(x^3 sqrt(2x^4 - 2x^2 + 1)`

Dividing numerator and denominator by x⁵,

I = `int ((1/x^3 - 1/x^5)dx)/sqrt(2 - 2/x^2 + 1/x^4)`

Put `2 - 2/x^2 + 1/x^4` = t

`\implies (4/x^3 - 4/x^5)dx` = dt

∴ I = `1/4 int dt/sqrt(t)`

= `1/2 sqrt(t) + C`

= `1/2 sqrt(2 - 2/x^2 + 1/x^4) + C`