Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{1}{\sqrt{x}+x} d x$

Solution:

$x=t^{2}$

$d(x)=2 t \cdot d t$

$d x=2 t \cdot d t$

Substituting $t$ and dt we get

$\Rightarrow \int \frac{2 t \cdot d t}{t^{2}+t}$

$\Rightarrow 2 \int \frac{t \cdot d t}{t^{2}+t}$

$\Rightarrow 2 \int \frac{1}{1+t} d t$

$\Rightarrow 2(\ln |1+t|)$

But $t=\sqrt{x}$

$\Rightarrow 2(\ln |1+\sqrt{x}|)+c$

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