I=∫dxx(x^5+3) I=∫x^4x^5(x^5+3)dx Put t=x^5 5x^4 dx=dt Therefore, I=∫dt5t(t+3) =15∫13[1t 1t+3]dt =115[log |t| log |t+3|]+C ...

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Integration Using Substitution

Evaluate : ∫...

Question

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

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Solution

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

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

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

$= \frac{1}{5} \int \frac{1}{3} [\frac{1}{t} - \frac{1}{t + 3}] d t$

$= \frac{1}{15} [log | t | - log | t + 3 |] + C$

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

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