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Question Number 97759 by bobhans last updated on 09/Jun/20
∫ ((sin^5 (x) dx)/( (√(cos (x))))) ?
$$\int\:\frac{\mathrm{sin}\:^{\mathrm{5}} \left({x}\right)\:{dx}}{\:\sqrt{\mathrm{cos}\:\left({x}\right)}}\:? \\ $$
Answered by bemath last updated on 09/Jun/20
∫ (((1−cos^2 x)^2 sin x dx)/( (√(cos x)))) =  set (√(cos x)) = z ⇒ −sin x dx = 2z dz  ∫ (((1−z^4 )^2 )/z) (−2z dz )=  −2∫ (1−2z^4 +z^8 ) dz =  −2z + (4/5)z^5 −(2/9)z^9 +c =  −2(√(cos x))+(4/5)cos^2 x(√(cos x)) −(2/9)cos^4 x (√(cos x)) + c
$$\int\:\frac{\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}\right)^{\mathrm{2}} \mathrm{sin}\:{x}\:{dx}}{\:\sqrt{\mathrm{cos}\:{x}}}\:= \\ $$$${set}\:\sqrt{\mathrm{cos}\:{x}}\:=\:{z}\:\Rightarrow\:−\mathrm{sin}\:{x}\:{dx}\:=\:\mathrm{2}{z}\:{dz} \\ $$$$\int\:\frac{\left(\mathrm{1}−{z}^{\mathrm{4}} \right)^{\mathrm{2}} }{{z}}\:\left(−\mathrm{2}{z}\:{dz}\:\right)= \\ $$$$−\mathrm{2}\int\:\left(\mathrm{1}−\mathrm{2}{z}^{\mathrm{4}} +{z}^{\mathrm{8}} \right)\:{dz}\:= \\ $$$$−\mathrm{2}{z}\:+\:\frac{\mathrm{4}}{\mathrm{5}}{z}^{\mathrm{5}} −\frac{\mathrm{2}}{\mathrm{9}}{z}^{\mathrm{9}} +{c}\:= \\ $$$$−\mathrm{2}\sqrt{\mathrm{cos}\:{x}}+\frac{\mathrm{4}}{\mathrm{5}}\mathrm{cos}\:^{\mathrm{2}} {x}\sqrt{\mathrm{cos}\:{x}}\:−\frac{\mathrm{2}}{\mathrm{9}}\mathrm{cos}\:^{\mathrm{4}} {x}\:\sqrt{\mathrm{cos}\:{x}}\:+\:{c}\: \\ $$
Commented by bobhans last updated on 09/Jun/20
alright
$$\mathrm{alright} \\ $$

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