Evaluate the following integrals:

Question:

Evaluate the following integrals: $\int \frac{1}{x^{1 / 3}\left(x^{1 / 3}-1\right)} d x$

Solution:

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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