bemath-1-x-2-9-4x-2-dx-

Question Number 106705 by bemath last updated on 06/Aug/20

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\int \frac{1 + x^{2}}{\sqrt{9 - {4 x}^{2}}} dx

Answered by mathmax by abdo last updated on 06/Aug/20

I = \int \frac{1 + x^{2}}{\sqrt{9 - {4 x}^{2}}} dx \Rightarrow I = \int \frac{1 + x^{2}}{3 \sqrt{1 - \frac{4}{9} x^{2}}} we do the changement

\frac{2}{3} x = sint \Rightarrow I = \int \frac{1 + \frac{9}{4} \sin^{2} t}{3 cost} \times \frac{3}{2} cost dt

= \frac{1}{2} \int \frac{4 + {9 \sin}^{2} t}{4} dt = \frac{1}{2} t + \frac{9}{8} \int \sin^{2} t dt

= \frac{t}{2} + \frac{9}{16} \int (1 - \cos (2 t)) dt = \frac{t}{2} + \frac{9 t}{16} - \frac{9}{32} \sin (2 t) + C

= \frac{17 t}{16} - \frac{9}{16} sint cost + C

= \frac{17}{16} \arcsin (\frac{2 x}{3}) - \frac{9}{16} . \frac{2 x}{3} \sqrt{1 - \frac{4}{9} x^{2}} + C

I = \frac{17}{16} \arcsin (\frac{2 x}{3}) - \frac{3 x}{8} \sqrt{1 - \frac{4}{9} x^{2}} + C