Evaluate the following intregals: 2x+1 x^2+2x-1 dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{2\text{x}+1}{\sqrt{\text{x}^2+2\text{x}-1}}\ \text{dx}$

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Answer

Let $\text{I}\int\frac{2\text{x}+1}{\sqrt{\text{x}^2+2\text{x}-1}}\ \text{dx}$Let $2\text{x}+1=\lambda\frac{\text{d}}{\text{dx}}(\text{x}^2+2\text{x}-1)+\mu$
$=\lambda(2\text{x}+2)+\mu$
$2\text{x}+1=(2\lambda)\text{x}+2\lambda+\mu$
Comapring the coefficient of like powers of x,
$2\lambda=2\Rightarrow\lambda=1$
$2\lambda+\mu=1\Rightarrow2(1)+\mu=1$
$\mu=-1$
So, $\text{I}=\int\frac{(2\text{x}+2)-1}{\sqrt{\text{x}^2+2\text{x}-1}}\text{dx}$
$\text{I}=\int\frac{(2\text{x}+2)}{\sqrt{\text{x}^2+2\text{x}-1}}\text{dx}-\int\frac{1}{\sqrt{\text{x}^2+2\text{x}+(1)^2-(1)^2-1}}$
$\text{I}=\int\frac{2\text{x}+2}{\sqrt{\text{x}^2+2\text{x}-1}}\text{dx}-\int\frac{1}{\sqrt{(\text{x}+1)^2-(\sqrt{2}^2})}$
$\text{I}=(2\sqrt{\text{x}^2+2\text{x}-1})-\log\big|(\text{x}+1)+\sqrt{(\text{x}+1)^2-(\sqrt{2}})^2\big|+\text{C}$ $\Big[\text{since},\int\frac{1}{\sqrt{\text{x}}}\text{dx}=2\sqrt{\text{x}}+\text{c},\int\frac{1}{\sqrt{\text{x}^2-\text{a}^2}}\text{dx}=\log\big|\text{x}+\sqrt{\text{x}^2-\text{a}^2}\big|+\text{C}\Big]$
$\text{I}=2\sqrt{\text{x}^2+2\text{x}-1}-\log\big|\text{x}+1+\sqrt{\text{x}^2+2\text{x}-1}\big|+\text{c}$

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