Evaluate: $\int \frac{\mathrm{x}+1}{\sqrt{2 \mathrm{x}+3}} \mathrm{dx}$


In these questions, little manipulation makes the questions easier to solve

Here multiply and divide by 2 we get

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

Add and subtract 1 from the numerator

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

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

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

$\Rightarrow \frac{1}{2}\left(\int \sqrt{2 x+3} d x-\int(2 x+3)^{\frac{-1}{2}} d x\right)$

$\Rightarrow \frac{1}{2} \times \frac{(2 x+3)^{\frac{3}{2}}}{2 \times \frac{2}{2}}-\frac{1}{2} \times \frac{(2 x+3)^{\frac{1}{2}}}{2 \times \frac{1}{2}}+c$

$\Rightarrow \frac{(2 x+3)^{\frac{3}{2}}}{6}-\frac{(2 x+3)^{\frac{1}{2}}}{2}+c$

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