Question

Please solve this ASAP
and please no fake answers ​

Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept}}}$

Here the Concept of integration has been used. Firstly we will simplify the given term. Then we will get two sub terms from the parent term. After that we will apply integration part by part in different terms. Then we will combine all the parts and thus we can get our answer.

Let's do it !!

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★ Solution :-

Given,

$\\\;\displaystyle{\bf{\mapsto\;\;\red{\int\:\dfrac{x\:e^{x}}{(1\;+\;x)^{2}}\:dx}}}$

This is the expression whose integral we have to find. This can be written as,

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\dfrac{(x\;+\;1\;-\;1)\:e^{x}}{(1\;+\;x)^{2}}\:dx}}$

Now this can be written as,

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\dfrac{e^{x}\:(1\;+\;x\;-\;1)}{(1\;+\;x)^{2}}\:dx}}$

$\\\;\displaystyle{\sf{\rightarrow\;\;\int\:\dfrac{e^{x}\:(1\;+\;x)}{(1\;+\;x)^{2}}\:dx\;-\;\int\:\dfrac{e^{x}\:\times\:1}{(1\;+\;x)^{2}}\:dx}}$

Now taking the common term, we get

$\\\;\displaystyle{\sf{\Longrightarrow\;\;\int\:e^{x}\:\bigg[\dfrac{(1\;+\;x)}{(1\;+\;x)^{2}}\;-\;\dfrac{1}{(1\;+\;x)^{2}}\bigg]\:dx}}$

This will give us,

$\\\;\displaystyle{\sf{\Longrightarrow\;\;\int\:e^{x}\:\bigg[\dfrac{1}{(1\;+\;x)}\;-\;\dfrac{1}{(1\;+\;x)^{2}}\bigg]\:dx}}$

• Let f(x) = 1/(1 + x)

• Let f'(x) = - 1/(1 + x)²

Now integrating both the terms part by part, we get

$\\\;\displaystyle{\sf{\Longrightarrow\;\;\int\:e^{x}\:\bigg[f\:(x)\;-\;(-\:f'\:(x))\bigg]\:dx}}$

$\\\;\displaystyle{\sf{\Longrightarrow\;\;e^{x}\:\int\:\bigg[f\:(x)\;+\;f'\:(x)\bigg]\:dx}}$

Now these terms can be written in the form of f(x).

$\\\;\displaystyle{\sf{\Longrightarrow\;\;\:e^{x}\;f\:(x)\;+\;C}}$

Now by applying the value of f(x), we get

$\\\;\displaystyle{\sf{\Longrightarrow\;\;\:e^{x}\;\times\;\dfrac{1}{1\;+\;x}\;+\;C}}$

$\\\;\displaystyle{\bf{\Longrightarrow\;\;\orange{\dfrac{e^{x}}{1\;+\;x}\;+\;C}}}$

Now this can be finally written as,

$\\\;\displaystyle{\bf{\Longrightarrow\;\;\int\:\dfrac{x\:e^{x}}{(1\;+\;x)^{2}}\:dx\;=\;\orange{\bigg(\dfrac{e^{x}}{1\;+\;x}\bigg)\;+\;C}}}$

This is our required answer.

$\\\;\underline{\boxed{\tt{Required\;Integral\;=\;\bf{\purple{\bigg(\dfrac{e^{x}}{1\;+\;x}\bigg)\;+\;C}}}}}$

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★ More to know :-

• Integration by parts ::

$\\\;\displaystyle{\tt{\leadsto\;\;\int\:u\:v\:dx\;=\;u\int\:v\:dx\;-\;\int\:u'(\int\:v\:dx)\:dx}}$

Answer 2

Answer:

Hope it will help you !!!!!!!!