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Question Number 32269 by abdo imad last updated on 22/Mar/18
find ∫  (x^3 /( (√(1+x^2 )))) dx
$${find}\:\int\:\:\frac{{x}^{\mathrm{3}} }{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$
Answered by mrW2 last updated on 22/Mar/18
∫  (x^3 /( (√(1+x^2 )))) dx  (1/2)∫  (x^2 /( (√(1+x^2 )))) dx^2   (1/2)∫  ((x^2 +1−1)/( (√(1+x^2 )))) dx^2   (1/2){∫ (√(x^2 +1)) dx^2 − ∫(1/( (√(1+x^2 )))) dx^2 }  (1/2){∫ (√(t+1)) dt− ∫(1/( (√(1+t)))) dt}  (1/2){(2/3)(t+1)(√(t+1))−2(√(t+1))}  {(1/3)(t+1)−1}(√(t+1))  (((t−2)(√(t+1)))/3)+C  (((x^2 −2)(√(x^2 +1)))/3)+C
$$\int\:\:\frac{{x}^{\mathrm{3}} }{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\:\:\frac{{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}^{\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\:\:\frac{{x}^{\mathrm{2}} +\mathrm{1}−\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}^{\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left\{\int\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}^{\mathrm{2}} −\:\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}^{\mathrm{2}} \right\} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left\{\int\:\sqrt{{t}+\mathrm{1}}\:{dt}−\:\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{t}}}\:{dt}\right\} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left\{\frac{\mathrm{2}}{\mathrm{3}}\left({t}+\mathrm{1}\right)\sqrt{{t}+\mathrm{1}}−\mathrm{2}\sqrt{{t}+\mathrm{1}}\right\} \\ $$$$\left\{\frac{\mathrm{1}}{\mathrm{3}}\left({t}+\mathrm{1}\right)−\mathrm{1}\right\}\sqrt{{t}+\mathrm{1}} \\ $$$$\frac{\left({t}−\mathrm{2}\right)\sqrt{{t}+\mathrm{1}}}{\mathrm{3}}+{C} \\ $$$$\frac{\left({x}^{\mathrm{2}} −\mathrm{2}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}{\mathrm{3}}+{C} \\ $$

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