Question Number 185181 by Noorzai last updated on 18/Jan/23
Commented by Shrinava last updated on 18/Jan/23
$$\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{lnx}\right)\right)\right)\right)\right)+\mathbb{C} \\ $$
Commented by Noorzai last updated on 18/Jan/23
$$? \\ $$
Commented by Gazella thomsonii last updated on 18/Jan/23
$$\mathrm{what}\:\mathrm{the}.... \\ $$
Answered by SEKRET last updated on 19/Jan/23
![[((ln(x)=a)),(((1/x)dx= da)) ] ∫(((((((1/a)/(ln(a) ))/(ln(ln(a))))/(ln(ln(ln(a)))))/(ln(ln(ln(ln(a))))))/(ln(ln(ln(ln(ln(a)))))))/(ln(ln(ln(ln(ln(ln(a)))))))) da [((ln(a)=b)),(((1/a)da=db)) ] ∫((((((1/b)/(ln(b)))/(ln(ln(b))))/(ln(ln(ln(b)))))/(ln(ln(ln(ln(b))))))/(ln(ln(ln(ln(ln(b))))))) db [((ln(b)=c )),(((1/b)db=dc)) ] ∫ (((((1/c)/(ln(c)))/(ln(ln(c)))))/(ln(ln(ln(c)))))/(ln(ln(ln(ln(c)))))) dc [((ln(c)= e)),(((1/c)dc= de)) ] ∫ ((((1/e)/(ln(e)))/(ln(ln(e))))/(ln(ln(ln(e))))) de [((ln(e)= f )),(((1/e) de = df)) ] ∫ (((1/f)/(ln(f)))/(ln(ln(f)))) df [((ln(f) = g )),(( (1/f) df = dg )) ] ∫ ((1/g)/(ln(g))) dg [((ln(g)=h )),(( (1/g)dg = h)) ] ∫ (1/h) dh =ln(h)+c h=ln(g)=ln(lnf))=ln(ln(ln(e)))= = ln(ln(ln(ln(c))))=ln(ln(ln(ln(ln(ln(ln(x)))))) answer ln(ln(ln(ln(ln(ln(ln(x)))))))+C ABDULAZIZ ABDUVALIYEV](Q185248.png)
$$\:\:\:\:\:\begin{bmatrix}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)=\boldsymbol{\mathrm{a}}}\\{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}}=\:\boldsymbol{\mathrm{da}}}\end{bmatrix} \\ $$$$\:\:\int\frac{\frac{\frac{\frac{\frac{\frac{\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{a}}\right)\:}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{a}}\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{a}}\right)\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{a}}\right)\right)\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{a}}\right)\right)\right)\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{a}}\right)\right)\right)\right)\right)\right)}\:\:\boldsymbol{\mathrm{da}}\:\:\begin{bmatrix}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{a}}\right)=\boldsymbol{\mathrm{b}}}\\{\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}}\boldsymbol{\mathrm{da}}=\boldsymbol{\mathrm{db}}}\end{bmatrix} \\ $$$$\int\frac{\frac{\frac{\frac{\frac{\frac{\mathrm{1}}{\boldsymbol{\mathrm{b}}}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{b}}\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{b}}\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{b}}\right)\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{b}}\right)\right)\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{b}}\right)\right)\right)\right)\right)}\:\:\boldsymbol{\mathrm{db}}\:\:\:\begin{bmatrix}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{b}}\right)=\boldsymbol{\mathrm{c}}\:}\\{\frac{\mathrm{1}}{\boldsymbol{\mathrm{b}}}\boldsymbol{\mathrm{db}}=\boldsymbol{\mathrm{dc}}}\end{bmatrix} \\ $$$$\:\int\:\frac{\frac{\frac{\frac{\frac{\mathrm{1}}{\boldsymbol{\mathrm{c}}}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{c}}\right)}}{\left.\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{c}}\right)\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{c}}\right)\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{c}}\right)\right)\right)\right)}\:\boldsymbol{\mathrm{dc}}\:\:\:\:\begin{bmatrix}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{c}}\right)=\:\boldsymbol{\mathrm{e}}}\\{\frac{\mathrm{1}}{\boldsymbol{\mathrm{c}}}\boldsymbol{\mathrm{dc}}=\:\boldsymbol{\mathrm{de}}}\end{bmatrix} \\ $$$$\int\:\frac{\frac{\frac{\frac{\mathrm{1}}{\boldsymbol{\mathrm{e}}}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{e}}\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{e}}\right)\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{e}}\right)\right)\right)}\:\boldsymbol{\mathrm{de}}\:\:\:\:\:\:\begin{bmatrix}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{e}}\right)=\:\boldsymbol{\mathrm{f}}\:\:}\\{\frac{\mathrm{1}}{\boldsymbol{\mathrm{e}}}\:\boldsymbol{\mathrm{de}}\:=\:\boldsymbol{\mathrm{df}}}\end{bmatrix} \\ $$$$\:\int\:\frac{\frac{\frac{\mathrm{1}}{\boldsymbol{\mathrm{f}}}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{f}}\right)}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{f}}\right)\right)}\:\:\boldsymbol{\mathrm{df}}\:\:\:\:\:\:\:\:\begin{bmatrix}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{f}}\right)\:=\:\boldsymbol{\mathrm{g}}\:\:}\\{\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{f}}}\:\boldsymbol{\mathrm{df}}\:=\:\boldsymbol{\mathrm{dg}}\:}\end{bmatrix} \\ $$$$\:\int\:\frac{\frac{\mathrm{1}}{\boldsymbol{\mathrm{g}}}}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{g}}\right)}\:\boldsymbol{\mathrm{dg}}\:\:\:\:\:\:\:\begin{bmatrix}{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{g}}\right)=\boldsymbol{\mathrm{h}}\:\:}\\{\:\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{g}}}\boldsymbol{\mathrm{dg}}\:=\:\boldsymbol{\mathrm{h}}}\end{bmatrix} \\ $$$$\:\int\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{h}}}\:\boldsymbol{\mathrm{dh}}\:=\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{h}}\right)+\boldsymbol{\mathrm{c}} \\ $$$$\left.\:\boldsymbol{\mathrm{h}}=\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{g}}\right)=\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{lnf}}\right)\right)=\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{e}}\right)\right)\right)= \\ $$$$=\:\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{c}}\right)\right)\right)\right)=\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)\right)\right)\right)\right)\right)\right. \\ $$$$\:\:\boldsymbol{\mathrm{answer}}\:\: \\ $$$$\:\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)\right)\right)\right)\right)\right)\right)+\boldsymbol{\mathrm{C}} \\ $$$$\boldsymbol{{ABDULAZIZ}}\:\:\:\boldsymbol{{ABDUVALIYEV}} \\ $$$$ \\ $$$$ \\ $$
Answered by aba last updated on 18/Jan/23
$$\mathrm{ln}\mid\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{ln}\left(\mathrm{x}\right)\right)\right)\right)\right)\mid+\mathrm{k}\:/\mathrm{k}\in\mathrm{R}\right.\right. \\ $$
Commented by Noorzai last updated on 18/Jan/23
$${Beexplained}\:{you} \\ $$
Commented by aba last updated on 18/Jan/23
$$\int\frac{\mathrm{u}^{'} }{\mathrm{u}}\mathrm{du}=\mathrm{ln}\mid\mathrm{u}\mid+\Bbbk \\ $$
Commented by Noorzai last updated on 19/Jan/23
$${thanks} \\ $$