Question-156729

Question Number 156729 by MathSh last updated on 14/Oct/21

Answered by MJS_new last updated on 15/Oct/21

\int \frac{x + 2}{(x^{2} + 3 x + 3) \sqrt{x + 1}} d x =

[t = \sqrt{x + 1} \to d x = 2 \sqrt{x + 1} d t]

= 2 \int \frac{t^{2} + 1}{t^{4} + t^{2} + 1} d t = \int \frac{d t}{t^{2} - t + 1} + \int \frac{d t}{t^{2} + t + 1} =

= \frac{2 \sqrt{3}}{3} \arctan \frac{\sqrt{3} (2 t - 1)}{3} + \frac{2 \sqrt{3}}{3} \arctan \frac{\sqrt{3} (2 t + 1)}{3} =

= \frac{2 \sqrt{3}}{3} (\arctan \frac{\sqrt{3} (2 \sqrt{x + 1} - 1)}{3} + \arctan \frac{\sqrt{3} (2 \sqrt{x + 1} + 1)}{3}) + C

Commented by MathSh last updated on 15/Oct/21

Very nice dear S er thankyou

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