Question

How to solve this question of integration

Answer 1

Answer :

We can solve this problem using by parts method.

Now,

$\int log (2 + {x}^{2}) dx$

$= log (2 + {x}^{2}) \int dx - \int \{ \frac{d}{dx} log (2 + {x}^{2}) \times \int dx \} dx + C$

where C is integral constant

$= x log (2 + {x}^{2}) - \int \frac{2x}{2 + {x}^{2}} x dx + C$

$= x log (2 + {x}^{2}) - 2 \int \frac{{x}^{2}}{2 + {x}^{2}} dx + C$

$= x log (2 + {x}^{2}) - 2 \int \frac {(2 + {x}^{2}) - 2}{2+{x}^{2}} dx + C$

$= x log (2 + {x}^{2}) - 2 \int dx + 4 \int \frac{dx}{{x}^{2} + {(\sqrt{2})}^{2}} + C$

$= x log (2 + {x}^{2}) - 2x + \frac{4}{\sqrt{2}} {tan}^{-1} \frac{x}{\sqrt{2}} + C$

$= x log (2 + {x}^{2}) - 2x + 2 \sqrt{2} {tan}^{-1} \frac{x}{\sqrt{2}} + C$

$\therefore \boxed{\int log (2 + {x}^{2}) dx = x log (2 + {x}^{2}) - 2x + 2 \sqrt{2} {tan}^{-1} \frac{x}{\sqrt{2}} + C}$

where C is integral constant

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