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Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
$\int\frac{2\text{x}+1}{\sqrt{3\text{x}+2}}\text{dx}$
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Answer

$\text{Let I}=\int\frac{2\text{x}+1}{\sqrt{3\text{x}+2}}\text{dx}$
Let $2\text{x}+1=\lambda(3\text{x}+2)+\mu$ On equating the coefficients of like powers of x on both sides, we get
$3\lambda=2\text{ and }2\lambda+\mu=1$
$\Rightarrow\lambda=\frac{2}{3}\text{ and }2\times\frac{2}{3}+\mu=1$
$\Rightarrow\lambda=\frac{2}{3}\text{ and }\mu=\frac{-1}{3}$
$\therefore\text{I}=\int\frac{\lambda(3\text{x}+2)+\mu}{\sqrt{3\text{x}+2}}\text{dx}$
$=\lambda\int\frac{3\text{x}+2}{\sqrt{3\text{x}+2}}\text{dx}+\mu\int\frac{1}{\sqrt{3\text{x}+2}}\text{dx}$
$=\lambda\int(3\text{x}+2)^\frac{1}{2}\text{dx}+\mu\int(3\text{x}+2)^\frac{-1}{2}\text{dx}$
$=\lambda\times\frac{(3\text{x}+2)^\frac{3}{2}}{\frac{3}{2}\times3}+\mu\frac{(3\text{x}+2)^\frac{1}{2}}{\frac{1}{2}\times3}\text{c}$
$=\frac{2}{3}\times\frac{2}{9}\times(3\text{x}+2)^\frac{3}{2}-\frac{1}{3}\times\frac{2}{3}(3\text{x}+2)^\frac{1}{2}+\text{c}$
$=\frac{4}{27}\times(3\text{x}+2)^\frac{3}{2}-\frac{2}{9}\times(3\text{x}+2)^\frac{1}{2}+\text{c}$
$=\frac{2}{9}\times\sqrt{3\text{x}+2}\Big[\frac{2}{3}\times(3\text{x}+2)-1\Big]+\text{c}$
$=\frac{2}{9}\times\sqrt{3\text{x}+2}\Big[\frac{6\text{x}+4-3}{3}\Big]+\text{c}$
$=\frac{2}{27}\times\sqrt{3\text{x}+2}(6\text{x}+1)+\text{c}$
$\therefore\text{I}=\frac{2}{27}\times(6\text{x}+1)\sqrt{3\text{x}+2}+\text{c}$

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