Question

Please give explanation to answer 23

Answer 1

$\int \: \frac{x {}^{2} + 1 }{ \sqrt{x {}^{2} + 2} } \: \: dx$

Add and subtract 1 in the given integral .

$\int \: \frac{x {}^{2} + 1 + 1 - 1 }{ \sqrt{x {}^{2} + 2} } \: \: dx \\ \\ = > \int \: \frac{x {}^{2} + 2 }{ \sqrt{x {}^{2} + 2} } \: \: dx \: - \int \: \frac{ 1 }{ \sqrt{x {}^{2} + 2} } \: \: dx \\ \\ = > \int \: \sqrt{x {}^{2} + 2 } \: \: dx \: - \int \: \frac{1}{ \sqrt{x {}^{2} + 2} } \: \: dx \:$

= ⅔ ( x² + 2 )^3/2 - 1/√2 tan^-1 x/√2 + C