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Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
$\int\frac{1}{\sqrt{\text{x}}+\sqrt[4]{\text{x}}}\text{dx}$
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Answer

Let $\text{I}=\int\frac{1}{\sqrt{\text{x}}+\sqrt[4]{\text{x}}}\text{dx}$

Let $\text{x}=\text{t}^{4}$

On differentiating both sides, we get

$\text{dx}=4\text{t}^{3}\text{dt}$

$\therefore\ \text{I}=\int\frac{4\text{t}^{3}}{\sqrt{\text{t}^{4}}+\sqrt[4]{\text{t}^{4}}}\text{dt}$

$=\int\frac{4\text{t}^3}{\text{t}^{2}+\text{t}}\text{dt}$

$=4\int\frac{\text{t}^{2}}{\text{t}+1}\text{dt}$

$4\int\frac{(\text{t}-1)(\text{t}+1)+1}{\text{t}+1}\text{dt}$

$=4\int\Big[(\text{t}-1)+\frac{1}{\text{t}+1}\Big]\text{dt}$

$=4\Big[\frac{\text{t}^2}{2}-\text{t}+\log(\text{t}+1)\big]+\text{C}$

$=2\sqrt{x}-4\sqrt[4]{\text{x}}+4\log\big(\sqrt[4]{\text{x}+1}\big)+\text{C}$

Hence, $\int\frac{1}{\sqrt{\text{x}}+\sqrt[4]{\text{x}}}\text{dx}=2\sqrt{x}-4\sqrt[4]{\text{x}}+4\log\big(\sqrt[4]{\text{x}+1}\big)+\text{C}$

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