int frac{sqrt{x^4+x^{-4}+2}}{x^3} d x= | vedclass

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Question

(A) Let $I = \int \frac{\sqrt{x^4+x^{-4}+2}}{x^3} d x$ We can rewrite the expression inside the square root as a perfect square: $x^4 + x^{-4} + 2 = (

Vedclass · Accepted Answer

$\log |x|-\frac{1}{4 x^4}+C$

Vedclass · Answer

(A) Let $I = \int \frac{\sqrt{x^4+x^{-4}+2}}{x^3} d x$ We can rewrite the expression inside the square root as a perfect square: $x^4 + x^{-4} + 2 = (x^2)^2 + (x^{-2})^2 + 2(x^2)(x^{-2}) = (x^2 + x^{-2})^2$ Substituting this into the integral: $I = \int \frac{\sqrt{(x^2 + x^{-2})^2}}{x^3} d x = \int \frac{x^2 + x^{-2}}{x^3} d x$ Now,divide each term in the numerator by $x^3$: $I = \int \left( \frac{x^2}{x^3} + \frac{x^{-2}}{x^3} \right) d x = \int \left( \frac{1}{x} + x^{-5} \right) d x$ Integrating term by term: $I = \int \frac{1}{x} d x + \int x^{-5} d x = \ln |x| + \frac{x^{-4}}{-4} + C$ $I = \ln |x| - \frac{1}{4x^4} + C$

$\int \frac{\sqrt{x^4+x^{-4}+2}}{x^3} d x=$

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