Evalute the following integrals: 1 x ( x +1) dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{1}{\sqrt{\text{x}}(\sqrt{\text{x}}+1)}\text{dx}$

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Answer

Let $\text{I}=\int\frac{1}{\sqrt{\text{x}}(\sqrt{\text{x}}+1)}\text{dx}$
Putting $\sqrt{\text{x}}+1=\text{t}$
$\Rightarrow\frac{1}{2\sqrt{\text{x}}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\frac{1}{\sqrt{\text{x}}}\text{dx}=2\text{dt}$
$\therefore\text{I}=2\int\frac{1}{\text{t}}\text{dt}$
$=2\text{ In }|\text{t}|+\text{C}$
$=2\text{ In }|\sqrt{\text{x}}+1|+\text{C }\big[\because\text{t}=\sqrt{\text{x}}+1\big]$

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