→∫√(2(x^2+32n+2))dx = ∫√(2)√((x+34)^2)+2 916 = ∫√(2)√((x+34)^2+2316) = ∫√(2)√((x+34)^2+(√(23)4)^2)dx = √(2) [x+3/42√((x+34)^2+(√(23)4)^2)+2316× 2 ...

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Differentiation of Inverse Trigonometric Functions

Solve √2x2+3...

Question

Solve
$\sqrt{2 x^{2} + 3 x + 4} d x$

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Solution

$\to \int \sqrt{2 (x^{2} + \frac{3}{2} n + 2)} d x = \int \sqrt{2} \sqrt{{(x + \frac{3}{4})}^{2}} + 2 - \frac{9}{16}$
$= \int \sqrt{2} \sqrt{{(x + \frac{3}{4})}^{2} + \frac{23}{16}} = \int \sqrt{2} \sqrt{{(x + \frac{3}{4})}^{2} + {(\frac{\sqrt{23}}{4})}^{2}} d x$
$= \sqrt{2} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x + 3 / 4}{2} \sqrt{{(x + \frac{3}{4})}^{2} + {(\frac{\sqrt{23}}{4})}^{2}} + \frac{23}{16 \times 2} s i n^{- 1} ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{x + \frac{3}{4}}{\frac{\sqrt{23}}{4}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
$= \sqrt{2} [\frac{4 x + 3}{8} \sqrt{x^{2} + \frac{3 x}{2} + 2} + \frac{23}{32} s i n^{- 1} (\frac{4 x + 3}{\sqrt{23}})] + c$

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