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xdx-x-4-please-




Question Number 206721 by ajfour last updated on 23/Apr/24
∫((xdx)/(x+4))=?       please
xdxx+4=?please
Answered by A5T last updated on 23/Apr/24
(x/(x+4))=((x+4−4)/(x+4))=1−(4/(x+4))  ⇒∫(x/(x+4))dx=∫1dx−4∫(1/(x+4))dx  =x−4ln∣x+4∣+c
xx+4=x+44x+4=14x+4xx+4dx=1dx41x+4dx=x4lnx+4+c
Commented by ajfour last updated on 23/Apr/24
Oh yes! thank you. this is the answer.
Ohyes!thankyou.thisistheanswer.
Commented by necx122 last updated on 23/Apr/24
It's funny how after after a long while we see Ajfour's post and it's a quite simple integral relative to what we know him for. Welcome back our prof.������
Commented by mr W last updated on 23/Apr/24
i guess it′s not him personally.
iguessitsnothimpersonally.
Commented by ajfour last updated on 23/Apr/24
yeah me, bit irritated yet amused by life. It will get okay, assure you.
Commented by mr W last updated on 23/Apr/24
then welcome back sir!
thenwelcomebacksir!
Answered by Ghisom last updated on 23/Apr/24
∫(x/(x+4))dx=       [t=((√(x+4))/2) → dx=4(√(x+4))dt]  =8∫(t−(1/t))dt=4t^2 −8ln t =  =x−4ln ∣x+4∣ +C
xx+4dx=[t=x+42dx=4x+4dt]=8(t1t)dt=4t28lnt==x4lnx+4+C
Commented by ajfour last updated on 23/Apr/24
wow!
wow!
Answered by BaliramKumar last updated on 24/Apr/24
put     x = 4tan^2 y               dx = 8tanysec^2 ydy  ∫((4tan^2 y)/(4tan^2 y+4))∙8tanysec^2 ydy  ∫((4tan^2 y)/(4sec^2 y))∙8tanysec^2 ydy  8∫tany(tan^2 y)dy = 8∫tany(sec^2 y−1)dy   8∫tanysec^2 ydy−8∫tanydy   8[((tan^2 y)/2)] − 8ln(secy)  4tan^2 y − 4ln(sec^2 y) = x − 4ln(((4sec^2 y)/4))  x − 4ln(((4tan^2 y + 4)/4)) = x − 4ln(((x + 4)/4))   x − 4ln∣x+4∣ + 4ln∣4∣  x − 4ln∣x+4∣ + C
putx=4tan2ydx=8tanysec2ydy4tan2y4tan2y+48tanysec2ydy4tan2y4sec2y8tanysec2ydy8tany(tan2y)dy=8tany(sec2y1)dy8tanysec2ydy8tanydy8[tan2y2]8ln(secy)4tan2y4ln(sec2y)=x4ln(4sec2y4)x4ln(4tan2y+44)=x4ln(x+44)x4lnx+4+4ln4x4lnx+4+C
Answered by peter frank last updated on 24/Apr/24
u=x+4  x=u−4  du=dx  ∫((xdu)/u)=∫((u−4)/u)du=∫1du−∫(4/u)=u−4ln u  ∫1du−∫(4/u)du=u−4ln u  x+4−4ln (x+4)
u=x+4x=u4du=dxxduu=u4udu=1du4u=u4lnu1du4udu=u4lnux+44ln(x+4)

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