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Question Number 105753 by bramlex last updated on 31/Jul/20

∫ ((x^2 +sin^2 x)/(x^2 +cos^2 x)) dx

$$\int\:\frac{{x}^{\mathrm{2}} +\mathrm{sin}\:^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} +\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx}\: \\ $$

Commented by Ar Brandon last updated on 31/Jul/20

No limits ?

$$\mathrm{No}\:\mathrm{limits}\:? \\ $$

Answered by bobhans last updated on 31/Jul/20

∫ ((x^2 +1−cos^2 x)/(x^2 +cos^2 x)) = ∫ ((x^2 +cos^2 x+(1−2cos^2 x))/(x^2 +cos^2 x))dx  = ∫ dx + ∫ ((1−2cos^2 x)/(x^2 +sin^2 x)) dx   = x + ∫ ((1−2cos^2 x)/(x^2 +sin^2 x)) dx   next

$$\int\:\frac{{x}^{\mathrm{2}} +\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} +\mathrm{cos}\:^{\mathrm{2}} {x}}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{cos}\:^{\mathrm{2}} {x}+\left(\mathrm{1}−\mathrm{2cos}\:^{\mathrm{2}} {x}\right)}{{x}^{\mathrm{2}} +\mathrm{cos}\:^{\mathrm{2}} {x}}{dx} \\ $$$$=\:\int\:{dx}\:+\:\int\:\frac{\mathrm{1}−\mathrm{2cos}\:^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} +\mathrm{sin}\:^{\mathrm{2}} {x}}\:{dx}\: \\ $$$$=\:{x}\:+\:\int\:\frac{\mathrm{1}−\mathrm{2cos}\:^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} +\mathrm{sin}\:^{\mathrm{2}} {x}}\:{dx}\: \\ $$$${next} \\ $$

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