Final answer to the problem
$\frac{1}{5}\ln\left|x\right|-\frac{1}{5}\ln\left|\sqrt{x^2+5}\right|+C_1$
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Step-by-step Solution
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- Integrate by partial fractions
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1
Rewrite the fraction $\frac{1}{x\left(x^2+5\right)}$ in $2$ simpler fractions using partial fraction decomposition
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$\frac{1}{5x}+\frac{-\frac{1}{5}x}{x^2+5}$
Learn how to solve problems step by step online. Find the integral int(1/(x(x^2+5)))dx. Rewrite the fraction \frac{1}{x\left(x^2+5\right)} in 2 simpler fractions using partial fraction decomposition. Simplify the expression. The integral \int\frac{1}{5x}dx results in: \frac{1}{5}\ln\left(x\right). The integral -\frac{1}{5}\int\frac{x}{x^2+5}dx results in: \frac{1}{5}\ln\left(\frac{\sqrt{5}}{\sqrt{x^2+5}}\right).
Final answer to the problem
$\frac{1}{5}\ln\left|x\right|-\frac{1}{5}\ln\left|\sqrt{x^2+5}\right|+C_1$
