∫ #160; #10; ln x ( 1 + ln x)^2 dx #10; #10;Put #160; #160; #10; ln x = t #160; #160; x = e^t #10; #10; #160; #10;dx = e ...

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Special Integrals - 1

∫ℓ nx/ 1 + ℓ ...

Question

$\int \frac{ℓ n x}{{(1 + ℓ n x)}^{2}} d x$

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Solution

$\int \frac{l n x}{(1 + l n x)^{2}} d x$

Put $l n x = t ⟹ x = e^{t}$

$⟹ d x = e^{t} d t$

$= \int \frac{t . e^{t}}{(1 + t)^{2}} d t$

$= \int e^{t} \times \frac{1}{(1 + t)} d t - \int e^{t} . \frac{1}{(1 + t)^{2}} d t$

$= \frac{1}{(1 + t)} e^{t} + \int \frac{1}{(1 + t)^{2}} e^{t} d t - \int e^{t} \frac{d t}{(1 + t)^{2}}$

$= \frac{e^{t}}{1 + t} + c$

$= \frac{x}{1 + l n x} + c$

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