Evaluate: dx x(x^ 5 +3) - Vidyadip

Question

Evaluate: $\int\frac{\text{dx}}{\text{x}\text{(x}^{5}\text{+3)}}$

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Answer

Let $I = \int\frac{\text{dx}}{\text{x}\text{(x}^{5}\text{+3)}}=\int\frac{\text{x}^{4}\text{dx}}{\text{x}^{5}\text{(x}^{5}\text{+3)}}=\frac{1}{5}\int\frac{\text{5x}^{4}\text{dx}}{\text{x}^{5}\text{(x}^{5}\text{+3)}}$ Let $x^5 = z$
$\Rightarrow 5x^4dx = dz$
$\therefore\text{I}=\frac{1}{5}\int\frac{\text{dz}}{\text{z(z+3)}}$
$=\frac{1}{5\times3}\int\frac{z+3-z}{z(z+3)}\text{dz}=\frac{1}{15}\int\frac{z+3}{z(z+3)}\text{dz}-\frac{1}{15}\frac{z}{z(z+3)}\text{dz}$
$=\frac{1}{15}\int.\frac{dz}{z}-\frac{1}{15}\int\frac{dz}{z+3}=\frac{1}{15}\left\{\text{log z}-\text{log }|z+3|\right\}+\text{C}$
$=\frac{1}{15}\log\Bigg|\frac{z}{z+3}\Bigg|+\text{C}=\frac{1}{15}\log\Bigg|\frac{x^5}{x^5+3}\Bigg|+\text{C}$

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