int frac{d x}{xleft(x^4+1right)}= | vedclass

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(D) To solve the integral $I = \int \frac{dx}{x(x^4+1)}$,multiply the numerator and denominator by $x^3$: $I = \int \frac{x^3 dx}{x^4(x^4+1)}$ Let $x^

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$\frac{1}{4} \log \left(\frac{x^4}{x^4+1}\right)+c$

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(D) To solve the integral $I = \int \frac{dx}{x(x^4+1)}$,multiply the numerator and denominator by $x^3$: $I = \int \frac{x^3 dx}{x^4(x^4+1)}$ Let $x^4 = t$,then $4x^3 dx = dt$,which implies $x^3 dx = \frac{dt}{4}$. Substituting these into the integral: $I = \frac{1}{4} \int \frac{dt}{t(t+1)}$ Using partial fractions,$\frac{1}{t(t+1)} = \frac{1}{t} - \frac{1}{t+1}$. $I = \frac{1}{4} \int \left( \frac{1}{t} - \frac{1}{t+1} \right) dt$ $I = \frac{1}{4} [\log|t| - \log|t+1|] + C$ $I = \frac{1}{4} \log \left| \frac{t}{t+1} \right| + C$ Substituting $t = x^4$ back: $I = \frac{1}{4} \log \left( \frac{x^4}{x^4+1} \right) + C$

$\int \frac{d x}{x\left(x^4+1\right)}=$

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