Question

Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.

Options

• 6.00

• 7.00

• 8.00

• 9.00

Answer 1

Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to 6.00.

Explanation:

Let I = `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4)x^4)`

= `int ((x - 4/x^5)dx)/(x^2 + 2/x^4)^(1/4)`  ...`((∵ int_α^βf(x)dx + int_a^bf^-1(x)dx = bβ - aα),("when"  a = f(α) and b = f(β)))`

Let `x^2 + 2/x^4` = t⁴

⇒ `(x - 4/x^5)dx` = 2t³dt

I = `2int (t^3dt)/t = (2t^3)/3`

= `2/3(x^6 + 2)^(3/4)/x^3 + C`

⇒ `n/(ℓm)` = 6