Evaluate the following intregals: 2x+5 x^2+2x+5 dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{2\text{x}+5}{\sqrt{\text{x}^2+2\text{x}+5}}\text{dx}$

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Answer

Let $\text{I}=\int\frac{2\text{x}+5}{\sqrt{\text{x}^2+2\text{x}+5}}\text{dx}$let $2\text{x}+5=\lambda\frac{\text{d}}{\text{dx}}(\text{x}^2+2\text{x}+5)+\mu$
$=\mu(2\text{x}+2)+\mu$
$2\text{x}+5=(2\lambda)\text{x}+2\lambda+\mu$
Compairing the coefficient of like powewrs of x,
$2\lambda=2\ \Rightarrow\lambda=1$
$2\lambda+\mu=5\ \Rightarrow 2(1)+\mu=5$
$\Rightarrow\mu=3$
So, $\text{I}=\int\frac{(2\text{x}+2)+3}{\sqrt{\text{x}^2+2\text{x}+5}}\text{dx}$
$=\int\frac{(2\text{x}+3)}{\sqrt{\text{x}^2+2\text{x}+5}}\text{dx}+3\int\frac{1}{\sqrt{\text{x}^2+2\text{x}+(1)^2-(1)^2+5}}\text{dx}$
$\text{I}=\int\frac{2\text{x}+3}{\sqrt{\text{x}^2+2\text{x}+5}}+3\int\frac{1}{\sqrt{(\text{x}+1)^2+(2)^2}}\text{dx}$
$\text{I}=2\sqrt{\text{x}^22\text{x}+5}+3\log\big|\text{x}+1+{\sqrt{(\text{x}+1)^2+(2)^2}}\big|+\text{C}$ $\big[\text{since}, \int\frac{1}{\text{x}}\text{dx}=2\sqrt{\text{x}+\text{C},\int\frac{1}{\sqrt{\text{x}^2+\text{a}^2}}}\text{dx}=\log\big|\text{x}+\sqrt{\text{x}^2+\text{a}^2}+\text{C}\big]$
$\text{I}=2\sqrt{\text{x}^2+2\text{x}+5}+3\log\big|\text{x}+1+\sqrt{\text{x}^2+2\text{x}+5}\big|+\text{C}$

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