∫$$1$$−$$x1+xdx$$ is equal to

Question

∫$$1$$−$$x1+xdx$$ is equal to

Infinity Learn · Accepted Answer

Let $$t=$$$$cos$$−$$1$$⁡x Now, ∫$$1$$−$$x1+xdx=$$∫$$1$$−$$cos$$⁡$$t1+$$$$cos$$⁡t⋅[$$2cos$$⁡$$t($$−$$sin$$⁡$$t)$$]dt⇒  $$I=$$−$$2$$∫$$2sin2$$⁡$$t/22cos2$$⁡$$t/2$$⋅$$2sin$$⁡$$t2$$⋅$$cos$$⁡$$t2$$⋅$$cos$$⁡tdt∵$$1$$−$$cos$$⁡$$x=2sin2$$⁡$$x2$$,$$1+$$$$cos$$⁡$$x=2cos2$$⁡$$x2and$$ $$sin$$⁡$$2t=2sin$$⁡tcos⁡t $$=$$−$$4$$∫$$sin2$$⁡$$t2$$⋅$$cos$$⁡$$tdt=$$−$$4$$∫$$1$$−$$cos$$⁡$$t2$$⋅$$cos$$⁡$$tdt=$$−$$2$$∫$$cos$$⁡t−$$cos2$$⁡$$tdt=$$−$$2$$∫$$cos$$⁡t−$$1+$$$$cos$$⁡$$2t2dt=$$−$$2sin$$⁡$$t+t+12sin$$⁡$$2t+C=$$−$$2sin$$⁡$$t+t+12$$×$$2sin$$⁡tcos⁡$$t+C=$$−$$21$$−$$cos2$$⁡$$t+t+122$$⋅$$1$$−$$cos2$$⁡t⋅$$cos$$⁡$$t+C=$$−$$21$$−$$x+$$$$cos$$−$$1$$⁡$$x+x$$⋅$$1$$−$$x+C=$$−$$21$$−$$x+$$$$cos$$−$$1$$⁡$$x+x$$−$$x2+C$$∴∫$$1$$−$$x1+xdx=$$−$$21$$−$$x+$$$$cos$$−$$1$$⁡$$x+x$$−$$x2+C$$

$\int \sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} dx$ is equal to

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∫1−x1+xdx is equal to

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$\int \sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} dx$ is equal to