Final answer:
To solve the integral ∫ dx/ x²( x⁴+ 1) ³/⁴, we can use a substitution. Let u = x⁴ + 1, then differentiate both sides to find du = 4x³ dx. Rearranging, we have dx = du / (4x³). Substituting these values back into the integral, we simplify it to (1/4) ∫ du / (x² * (x⁴ + 1)^(³/⁴)).
Step-by-step explanation:
Integral of ∫ dx/ x²( x⁴+ 1) ³/⁴
To solve this integral, we can use a substitution. Let u = x⁴ + 1, then differentiate both sides to find du = 4x³ dx. Rearranging, we have dx = du / (4x³). Substituting these values back into the integral, we get:
∫ dx/ x²( x⁴+ 1) ³/⁴ = ∫ du / (x² * u^(³/⁴)) = (1/4) ∫ du / (x² * u^(³/⁴))
Now, notice that the integral on the right side is easier to solve. Let's simplify it further:
(1/4) ∫ du / (x² * u^(³/⁴)) = (1/4) ∫ du / (x² * (x⁴ + 1)^(³/⁴))
This is the final answer to the integral.