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dx-1-x-1-4-x-




Question Number 157977 by tounghoungko last updated on 30/Oct/21
∫ (dx/((1+(x)^(1/4)  )(√x))) =?
dx(1+x4)x=?
Answered by puissant last updated on 30/Oct/21
Ω=∫(dx/((1+(x)^(1/4) )(√x)))  u=(√(x )) → du=(1/(2(√x)))dx → dx= 2(√x)du  ⇒ Ω = ∫((2(√x)du)/((1+(√u))(√x))) = 2∫(du/((1+(√u))))  t=(√u) → t^2 =u → 2tdt=du  ⇒ Ω = 2∫((2tdt)/((1+t))) = 4∫((t+1−1)/((1+t)))dt  ⇒ Ω = 4∫dt−4∫(1/((1+t)))dt  ⇒ Ω = 4t−4ln∣1+t∣+C    ⇒ Ω = 4(√u)−4ln∣1+(√u)∣+C              ∴∵  Ω = 4((x)^(1/4)  − ln∣1+(x)^(1/4) ∣+C             ............Le puissant............
Ω=dx(1+x4)xu=xdu=12xdxdx=2xduΩ=2xdu(1+u)x=2du(1+u)t=ut2=u2tdt=duΩ=22tdt(1+t)=4t+11(1+t)dtΩ=4dt41(1+t)dtΩ=4t4ln1+t+CΩ=4u4ln1+u+C∴∵Ω=4(x4ln1+x4+CLepuissant

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