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Indefinite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration2 Marks
Question
Evaluate the following : $\int \frac{1}{\sqrt{3 x+1}-\sqrt{3 x-5}} \cdot d x$
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Answer

$\int \frac{1}{\sqrt{3 x+1}-\sqrt{3 x-5}} \cdot d x$
$
\begin{aligned}
= & \int\left(\frac{1}{\sqrt{3 x+1}-\sqrt{3 x-5}}\right) \cdot\left(\frac{\sqrt{3 x+1}+\sqrt{3 x-5}}{\sqrt{3 x+1}+\sqrt{3 x-5}}\right) \cdot d x \\
& =\int \frac{\sqrt{3 x+1}+\sqrt{3 x-5}}{3 x+1-3 x+5} \cdot d x \\
& =\int \frac{\sqrt{3 x+1}+\sqrt{3 x-5}}{6} \cdot d x \\
& =\frac{1}{6} \cdot \int\left((3 x+1)^{\frac{1}{2}}+(3 x-5)^{\frac{1}{2}}\right) \cdot d x \\
& =\frac{1}{6} \cdot\left\{\int(3 x+1)^{\frac{1}{2}} \cdot d x+\int(3 x-5)^{\frac{1}{2}} \cdot d x\right\} \\
& =\frac{1}{6} \cdot\left\{\frac{(3 x+1)^{\frac{1}{2}}+1}{\left(\left(\frac{1}{2}+1\right) \cdot 3\right.}+\frac{(3 x-5)^{\frac{1}{2}+1}}{\left(\frac{1}{2}+1\right) \cdot 3}\right\}+c \\
& =\frac{1}{18} \cdot\left\{\frac{2}{3}(3 x+1)^{\frac{3}{2}}+\frac{2}{3}(3 x-5)^{\frac{3}{2}}\right\}+c \\
& =\frac{1}{27} \cdot\left\{(3 x+1)^{\frac{3}{2}}+(3 x-5)^{\frac{3}{2}}\right\}+c
\end{aligned}
$

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