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Indefinite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration2 Marks
Question
Evaluate the following : $\int \frac{2 x-7}{\sqrt{3 x-2}} d x$
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Answer

Express $(2 x-7)$ in terms of $(3 x-2)$
$
\begin{aligned}
& 2 x-7=\frac{2}{3}(3 x-2)+\frac{4}{3}-7 \\
& =\frac{2}{3}(3 x-2)-\frac{17}{3} \\
& I=\int\left[\frac{\frac{2}{3}(3 x-2)-\frac{17}{3}}{\sqrt{3 x-2}}\right] \cdot d x \\
& =\int\left[\frac{\frac{2}{3}(3 x-2)}{\sqrt{3 x-2}}-\frac{\frac{17}{3}}{\sqrt{3 x-2}}\right] \cdot d x \\
& =\frac{2}{3} \int \sqrt{3 x-2} \cdot d x-\frac{17}{3} \int \frac{1}{\sqrt{3 x-2}} \cdot d x \\
& =\frac{2}{3} \int(3 x-2)^{\frac{1}{2}} \cdot d x-\frac{17}{3} \int \frac{1}{\sqrt{3 x-2}} \cdot d x \\
& =\frac{2}{3} \cdot \frac{(3 x-2)^{\frac{3}{2}}}{\left(\frac{3}{2}\right)} \cdot \frac{1}{3}-\frac{17}{3} \cdot 2 \cdot(\sqrt{3 x-2}) \cdot \frac{1}{3}+c \\
& =\frac{4}{27} \cdot(3 x-2)^{\frac{3}{2}}-\frac{34}{9} \cdot(3 x-2)^{\frac{1}{2}}+c \\
&
\end{aligned}
$

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