calculate-1-1-x-2-3-1-x-1-x-dx-

Question Number 59168 by maxmathsup by imad last updated on 05/May/19

c a l c u l a t e \int_{- 1}^{1} \frac{x^{2} + 3}{\sqrt{1 + x} + \sqrt{1 - x}} d x

Answered by ajfour last updated on 05/May/19

\frac{I}{2} = \int_{0}^{1} \frac{x^{2} + 3}{\sqrt{1 + x} + \sqrt{1 - x}} dx

= \int_{0}^{1} \frac{(x^{2} + 3) (\sqrt{1 + x} - \sqrt{1 - x})}{2 x} dx

I = \int_{0}^{1} xdx - 3 \int_{0}^{\frac{π}{2}} \frac{\sqrt{2} \cos θ - \sqrt{2} \sin θ}{(\cos^{2} θ - \sin^{2} θ)} (- 4 \sin θ \cos θ d θ)

= \frac{1}{2} + 12 \sqrt{2} \int_{0}^{\frac{π}{2}} \frac{\sin θ \cos θ}{\cos θ + \sin θ} d θ

= \frac{1}{2} + 24 \sqrt{2} \int_{0}^{π / 4} \frac{\sin θ d θ}{1 + \tan θ}

\dots \dots

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