#160; #160; #160; #160; #160; #8747;2x2x+12dx= #8747;2x+1 12x+12dx= #8747;2x+12x+12 12x+12dx= #8747;dx2x+1 #8747;2x+1 2dx=log2x+12 2x+1 2+ ...

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Special Integrals - 2

∫ 2 x 2 x+12 ...

Question

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

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Solution

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

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