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Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals3 Marks
Question
$\int\frac{2\text{x}}{(2\text{x}+1)^2}\text{dx}$
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Answer

$\int\frac{2\text{x}}{(2\text{x}+1)^2}\text{dx}$
$=\int\bigg(\frac{2\text{x}+1-1}{(2\text{x}+1)^2}\bigg)\text{dx}$
$=\int\bigg[\frac{2\text{x}+1}{(2\text{x}+1)^2}-\frac{1}{(2\text{x}+1)^2}\bigg]\text{dx}$
$=\int\frac{\text{dx}}{2\text{x}+1}-\int(2\text{x}+1)^{-2}\text{dx}$
$=\frac{\log(2\text{x}+1)}{2}-\bigg[\frac{(2\text{x}+1)^{-2+1}}{2(-2+1)}\bigg]+\text{c}$
$=\frac{\log(2\text{x}+1)}{2}+\frac{(2\text{x}+1)^{-1}}{2}+\text{c}$
$=\frac{\log(2\text{x}+1)}{2}+\frac{1}{2(2\text{x}+1)}+\text{c}$

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