Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Integration (p-1)2 Marks
Question
Evalute : $\int \frac{1}{\sqrt{x}+x} d x$
✓
Answer
$ \text { Let } \begin{aligned} I & =\int \frac{1}{\sqrt{x}+x} d x=\int \frac{1}{\sqrt{x}(1+\sqrt{x})} d x \\ & =\int \frac{1}{1+\sqrt{x}} \cdot \frac{1}{\sqrt{x}} d x \end{aligned} $ Put $1+\sqrt{x}=t \quad \therefore \frac{1}{2 \sqrt{x}} d x=d t$ $ \begin{aligned} \therefore & \frac{1}{\sqrt{x}} d x=2 d t \\ \therefore I & =\int \frac{1}{t} \cdot 2 d t=2 \int \frac{1}{t} d t \\ & =2 \log |t|+c=2 \log |1+\sqrt{x}|+c . \end{aligned} $
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