d x x ( x 5 + 3 ) is equal to:

Question

$\int \frac{d x}{x (x^{5} + 3)}$ is equal to:

Smart Achievers · Accepted Answer

(2)

Ans. $\frac{1}{15} \log$ $| \frac{x^{5}}{x^{5} + 3} |$ $+ C$

Explanation:

$Let$ $I = \int \frac{d x}{x (x^{5} + 3)}$

Multiply by $x^{4}$ in numerator and denominator, we get,

$I = \int \frac{x^{4}}{x^{5} (x^{5} + 3)} d x$

$Let$ $x^{5} = t$

$5 x^{4} d x = d t$

$I = \int \frac{1}{5 t (t + 3)} d t$

$= \frac{1}{5} \times \frac{1}{3} \int \frac{t + 3 - t}{t (t + 3)} d t$

$= \frac{1}{15} [\int \frac{1}{t} d t - \int \frac{1}{t + 3} d t]$

$= \frac{1}{15} [\log t - \log (t + 3)]$

$I = \frac{1}{15} \log$ $| \frac{x^{5}}{x^{5} + 3} |$ $+ C$

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