Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals3 Marks
Question
Integrate the function $\frac{1}{\sqrt{x^{2}+2 x+2}}$
✓
Answer
Clearly, $\int \frac{1}{\sqrt{x^{2}+2 x+2}} d x=\int \frac{1}{\sqrt{(x+1)^{2}+(1)^{2}}} d x$ Let x + 1 = t $\Rightarrow$ dx = dt $\Rightarrow \int \frac{1}{\sqrt{(x+1)^{2}+(1)^{2}}} d x=\int \frac{1}{\sqrt{t^{2}+1}} d t$ $=\log |t+\sqrt{t^{2}+1}|+C$ $=\log |(x+1)+\sqrt{(x+1)^{2}+1}|+C$ $=\log |(x+1)+\sqrt{x^{2}+2 x+2}|+C$
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