Question

\[\int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx\]

Answer 1

\[\text{ Let  I } = \int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx\]
\[ \text{ Let  x }= t^4 \]
\[ \text{On differentiating both sides, we get}\]
\[ dx = 4 t^3 dt\]
\[ \therefore I = \int\frac{4 t^3}{\sqrt{t^4} + \sqrt[4]{t^4}}dt\]
\[ = \int\frac{4 t^3}{t^2 + t}dt\]
\[ = 4\int\frac{t^2}{t + 1}dt\]

\[ = 4\int\frac{\left( t - 1 \right)\left( t + 1 \right) + 1}{t + 1}dt\]
\[ = 4\int\left[ \left( t - 1 \right) + \frac{1}{t + 1} \right]dt\]
\[ = 4\left[ \frac{t^2}{2} - t + \log\left( t + 1 \right) \right] + c\]
\[ = 2\sqrt{x} - 4\sqrt[4]{x} + 4 \log\left( \sqrt[4]{x} + 1 \right) + c\]
\[Hence, \int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx = 2\sqrt{x} - 4\sqrt[4]{x} + 4 \log\left( \sqrt[4]{x} + 1 \right) + c\]