Question

Do Integration please
$\huge \red \: \green \int \blue (\frac{ \sqrt[]{x} + 1}{x} \blue)^{2}$
​

Answer 1

Step-by-step explanation:

Given:

$\int \bigg(\frac{ \sqrt[]{x} + 1}{x} \bigg)^{2}dx$

To find: Solution of Integration

Solution:

First expand the given function

$( {a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab$

So,

$\int \frac{ (\sqrt[]{x} + 1)^{2} }{x ^{2} } dx$

$\int \frac{ (\sqrt{x}) ^{2} + 1 + 2 \sqrt{x} }{x ^{2} } dx \\ \\ \int \frac{ x+ 1 + 2 \sqrt{x} }{x ^{2} } dx \\ \\ \int \bigg( \frac{ x}{ {x}^{2}} + \frac{1}{ {x}^{2} } + \frac{2 \sqrt{x} }{ {x}^{2} } \bigg) dx \\ \\ \int \frac{ 1}{ {x}} dx + \int \frac{1}{ {x}^{2} } dx + \int\frac{2 \sqrt{x} }{ {x}^{2} } dx \\ \\ \int \frac{ 1}{ {x}} dx + \int \frac{1}{ {x}^{2} } dx + 2 \int {x}^{ \frac{ - 3}{2} } dx$

Now,do integration by applying power rule

$log \: x + \frac{ {x}^{ - 2 + 1} }{ - 2 + 1} + 2 \frac{ {x}^{ \frac{ - 3}{2} + 1 } }{ \frac{ - 3}{2} + 1} + C$

$log \: x - {x}^{ - 1} - 4 {x}^{ \frac{ - 1}{2} } + C$

or

$log \: x - \frac{1}{x} - \frac{4}{ \sqrt{x} } + C$

Thus,

Final Answer:

$\int \bigg(\frac{ \sqrt[]{x} + 1}{x} \bigg)^{2}dx =log \: x - \frac{1}{x} - \frac{4}{ \sqrt{x} } + C$

Hope it helps you.

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