Evaluate:

Question:

Evaluate: $\int \frac{2 \mathrm{x}}{(2 \mathrm{x}+1)^{2}} \mathrm{dx}$

Solution:

Let $I=\int \frac{2 x}{(2 x+1)^{2}} d x$

$=\int \frac{2 x+1-1}{(2 x+1)^{2}} d x$

$=\int \frac{2 \mathrm{x}+1}{(2 \mathrm{x}+1)^{2}}-\frac{1}{(2 \mathrm{x}+1)^{2}} \mathrm{dx}$

$=\int \frac{1}{(2 \mathrm{x}+1)}-(2 \mathrm{x}+1)^{-2} \mathrm{dx}$

$=\frac{1}{2} \log |2 \mathrm{x}+1|-\frac{(2 \mathrm{x}+1)^{-2+1}}{-2+1(2)}$

$=\frac{1}{2} \log |2 \mathrm{x}+1|-\frac{(2 \mathrm{x}+1)^{-1}}{-2}$

Hence, $I=\frac{1}{2} \log |2 x+1|+\frac{1}{2(2 x+1)}+C$

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