Verify the following: 2x-1 2x+3 dx=x- |(2x+3)^3 |+c - Vidyadip

Question

Verify the following:
$\int\frac{2\text{x}-1}{2\text{x}+3}\text{dx}=\text{x}-\log\big|(2\text{x}+3)^3\big|+\text{c}$

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Answer

Let $\int\frac{2\text{x}-1}{2\text{x}+3}\text{dx}$
$=\int\frac{2\text{x}+3-2}{2\text{x}+3}\text{dx}$
$=\int\Big(\frac{2\text{x}+3}{2\text{x}+3}-\frac{2}{2\text{x}+3}\Big)\text{dx}$
$=\int1\text{dx}-4\int\frac{1}{2\text{x}+3}\text{dx}$
$=\text{x}-\frac{4}{2}\log\big|(2\text{x}+3)\big|+\text{C}$ $\Big[\because\int\frac{1}{\text{ax}+\text{b}}\text{dx}=\frac{1}{\text{a}}\log|\text{ax}+\text{b}|\Big]$
$=\text{x}-2\log|(2\text{x}+3)|+\text{C}$
$=\text{x}-\log|(2\text{x}+3)^2|+\text{C}$ $\big[\because\ \text{a}\log\text{b}=\log\text{b}^\text{a}\big]$

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