#8747;x+12x+3dx=12 #8747;2x+22x+3dx=12 #8747;2x+3 12x+3dx=12 #8747;2x+32x+3 12x+3dx=12 #8747;2x+3 12x+3dx=12 #8747;2x+312dx #8747;2x+3 12dx=122 ...

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Properties of Inequalities

∫ x+12 x+3 d ...

Question

$\int \frac{x + 1}{\sqrt{2 x + 3}} d x$

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Solution

$\int (\frac{x + 1}{\sqrt{2 x + 3}}) d x = \frac{1}{2} \int (\frac{2 x + 2}{\sqrt{2 x + 3}}) d x = \frac{1}{2} \int (\frac{2 x + 3 - 1}{\sqrt{2 x + 3}}) d x = \frac{1}{2} \int (\frac{2 x + 3}{\sqrt{2 x + 3}} - \frac{1}{\sqrt{2 x + 3}}) d x = \frac{1}{2} \int (\sqrt{2 x + 3} - \frac{1}{\sqrt{2 x + 3}}) d x = \frac{1}{2} [\int {(2 x + 3)}^{\frac{1}{2}} d x - \int {(2 x + 3)}^{- \frac{1}{2}} d x] = \frac{1}{2} [\frac{{(2 x + 3)}^{\frac{1}{2} + 1}}{2 (\frac{1}{2} + 1)} - \frac{{(2 x + 3)}^{- \frac{1}{2} + 1}}{2 (- \frac{1}{2} + 1)} + C] = \frac{1}{2} [\frac{1}{3} {(2 x + 3)}^{\frac{3}{2}} - {(2 x + 3)}^{\frac{1}{2}} + C] = \frac{1}{6} {(2 x + 3)}^{\frac{3}{2}} - \frac{1}{2} {(2 x + 3)}^{\frac{1}{2}} + C$

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