P-rove-that-dx-b-4-2ax-2-c-tan-1-2-a-x-c-b-4-2-a-c-b-4-C-if-a-c-b-4-gt-0-

Question Number 202388 by Calculusboy last updated on 25/Dec/23

P r o v e t h a t : \int \frac{d x}{b^{4} + 2 {a x}^{2} + c} = \frac{{t a n}^{- 1} (\frac{\sqrt{2} \sqrt{a} x}{\sqrt{c + b^{4}}})}{\sqrt{2} \sqrt{a} \sqrt{c + b^{4}}} + C

i f a \cdot (c + b^{4}) > 0

Answered by witcher3 last updated on 26/Dec/23

\int \frac{dx}{b^{4} + c + {2 ax}^{2}} = \int \frac{dx}{2 a (x^{2} + \frac{b^{4} + c}{2 a})}; β^{2} = \frac{b^{4} + c}{2 a} > 0

= \frac{1}{2 a} \int \frac{dx}{x^{2} + β^{2}} = \frac{1}{2 a β} \tan^{- 1} (\frac{x}{β}) + k; k \in R

= \frac{1}{2 a \sqrt{\frac{b^{4} + c}{2 a}}} \tan^{- 1} (x \sqrt{\frac{2 a}{b^{4} + c}}) + k

= \frac{1}{\sqrt{2} . \sqrt{a} . \sqrt{b^{4} + c}} . \tan^{- 1} (\frac{x \sqrt{2} . \sqrt{a}}{\sqrt{c + b^{4}}}) + k

Commented by Calculusboy last updated on 26/Dec/23

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