Question

integrate x/(x+1)² dx​

Answer 1

the solution is in attachment.

Answer 2

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:\displaystyle\int\tt \dfrac{x}{ {(x + 1)}^{2} }dx$

☆ On adding and Subtracting 1 in numerator, we get

$\rm = \: \: \:\displaystyle\int\tt \dfrac{x + 1 - 1}{ {(x + 1)}^{2} }dx$

$\rm = \: \: \:\displaystyle\int\tt \dfrac{x + 1}{ {(x + 1)}^{2} }dx + \displaystyle\int\tt \dfrac{1}{ {(x + 1)}^{2} }dx$

$\rm = \: \: \:\displaystyle\int\tt \dfrac{1}{ {(x + 1)}}dx + \displaystyle\int\tt {(x + 1)}^{ - 2} dx$

$\: \rm= \: \: \: log(x + 1) + \dfrac{ {(x + 1)}^{ - 2 + 1} }{ - 2 + 1} + c$

$\boxed{ \sf \:\because \: \displaystyle\int\tt \dfrac{1}{x}dx = log(x) +c \: \:\: and \: \displaystyle\int\tt {x}^{n}dx = \dfrac{ {x}^{n + 1}}{n + 1}+c}$

$\: \rm= \: \: \: log(x + 1) + \dfrac{ {(x + 1)}^{ -1} }{ -1} + c$

$\: \rm= \: \: \: log(x + 1) - \dfrac{1}{x + 1} + c$

Additional Information :-

$\green{\boxed{ \bf \:\int {x}^{n} \: dx = \dfrac{ {x}^{n + 1} }{n + 1} + c}}$

$\green{\boxed{ \bf \:\int \: kdx =kx + c}}$

$\green{\boxed{ \bf \:\int \: cosx \: dx =sinx + c}}$

$\green{\boxed{ \bf \:\int \: sinx \: dx = - \: cosx + c}}$

$\green{\boxed{ \bf \: \int \: \dfrac{1}{x} dx = log(x) + c}}$

$\green{\boxed{ \bf \: \int \: secx \: tanx \: dx =secx + c}}$

$\green{\boxed{ \bf \: \int \: cosecx \: cotx \: dx = - \: cosecx + c}}$

$\green{\boxed{ \bf \: \int \: {sec}^{2}x \: dx = tanx+ c}}$

$\green{\boxed{ \bf \: \int \: {cosec}^{2}x \: dx = - \: cotx+ c}}$

$\green{\boxed{ \bf \: \int \: cotx \: dx = log(sinx) + c}}$

$\green{\boxed{ \bf \: \int \: tanx \: dx = log(secx) + c}}$