int frac{5 x^2+3}{x^2left(x^2-2right)} d x= | vedclass

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(C) Let $I = \int \frac{5x^2+3}{x^2(x^2-2)} dx$. Using partial fractions,let $\frac{5x^2+3}{x^2(x^2-2)} = \frac{A}{x^2} + \frac{B}{x^2-2}$. Then $5x^2

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$\frac{13}{4 \sqrt{2}} \log \left|\frac{x-\sqrt{2}}{x+\sqrt{2}}\right|+\frac{3}{2 x}+C$

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(C) Let $I = \int \frac{5x^2+3}{x^2(x^2-2)} dx$. Using partial fractions,let $\frac{5x^2+3}{x^2(x^2-2)} = \frac{A}{x^2} + \frac{B}{x^2-2}$. Then $5x^2+3 = A(x^2-2) + Bx^2$. Comparing coefficients of $x^2$: $A+B = 5$. Comparing constant terms: $-2A = 3 \implies A = -\frac{3}{2}$. Substituting $A$ in $A+B=5$: $-\frac{3}{2} + B = 5 \implies B = 5 + \frac{3}{2} = \frac{13}{2}$. Thus,$I = \int \left( -\frac{3}{2x^2} + \frac{13}{2(x^2-2)} \right) dx$. $I = -\frac{3}{2} \int x^{-2} dx + \frac{13}{2} \int \frac{1}{x^2-(\sqrt{2})^2} dx$. Using the formula $\int \frac{1}{x^2-a^2} dx = \frac{1}{2a} \log \left| \frac{x-a}{x+a} \right| + C$: $I = -\frac{3}{2} \left( -\frac{1}{x} \right) + \frac{13}{2} \cdot \frac{1}{2\sqrt{2}} \log \left| \frac{x-\sqrt{2}}{x+\sqrt{2}} \right| + C$. $I = \frac{3}{2x} + \frac{13}{4\sqrt{2}} \log \left| \frac{x-\sqrt{2}}{x+\sqrt{2}} \right| + C$.

$\int \frac{5 x^2+3}{x^2\left(x^2-2\right)} d x=$

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