Evaluate the following integrals: $\int \frac{1}{x^{1 / 3}\left(x^{1 / 3}-1\right)} d x$
Let I $=\int \frac{1}{x^{\frac{1}{3}}\left(x^{\left.\frac{1}{3}-1\right)}\right.} d x$
Multiplying and dividing by $x^{\frac{1}{3}}$
$\Rightarrow I=\int \frac{x^{\frac{1}{3}}}{x^{\frac{2}{3}}\left(x^{\frac{1}{3}}-1\right)} d x$
Let, $x^{\frac{1}{3}}-1=t \Rightarrow \frac{1}{3} x^{-\frac{2}{3}} d x=d t$
So, $\Rightarrow I=3 \int \frac{(t+1)}{t} d t$
$\Rightarrow I=3 \int\left(t+\frac{1}{t}\right) d t$
$\Rightarrow I=3\left(\frac{t^{2}}{2}+\log |t|\right)+c$
$\Rightarrow I=3\left(\frac{\left(x^{\frac{1}{3}}-1\right)^{2}}{2}+\log \left|\left(x^{\frac{1}{3}}-1\right)\right|\right)+c$
Therefore, $\int \frac{1}{\sqrt{x}+\sqrt[4]{x}} d x=3\left(\frac{\left(x^{\frac{1}{3}}-1\right)^{2}}{2}+\log \left|\left(x^{\frac{1}{3}}-1\right)\right|\right)+c$
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