First slide

Methods of integration

Question

The value of $\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}} is$

Moderate

A

$3 \sqrt{x} + 3 (\sqrt[3]{x}) - 6 \sqrt[6]{x} + 6 \log (\sqrt[6]{x} + 1) + C$
B

$2 \sqrt{x} + 6 (\sqrt[6]{x}) - 6 \log (\sqrt[6]{x} + 1) + C$
C

$2 \sqrt{x} - 3 (\sqrt[3]{x}) + 6 (\sqrt[6]{x}) - 6 \log (\sqrt[6]{x} + 1) + C$
D

None of the above

Solution

$Let I = \int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}$
Now let $x = t^{6} \Rightarrow dx = 6 t^{5} dt$
$\begin{array}{l} \Rightarrow I = \int \frac{6 t^{0}}{t^{3} + t^{2}} dt = 6 \int \frac{t^{3}}{t + 1} dt \\ = 6 \int (t^{2} - t + 1 - \frac{1}{t + 1}) dt \\ = 2 t^{3} - 3 t^{2} + 6 t - 6 \log (t + 1) + C \\ = 2 \sqrt{x} - 3 (\sqrt[3]{x}) + 6 (\sqrt[6]{x}) - 6 \log (\sqrt[6]{x} + 1) + C \end{array}$

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