Mathematics

# Evaluate the following integral:$\int { \cfrac { 1 }{ 1+\sqrt { x } } } dx$

##### SOLUTION
$\displaystyle \int \dfrac{1}{1+\sqrt{x}}dx$

Multiplying with root $x$

$\displaystyle \int\dfrac{1}{\sqrt{x}} \dfrac{\sqrt{x}}{1+\sqrt{x}}dx$

$\displaystyle \int\dfrac{1}{\sqrt{x}} \left[1- \dfrac{1}{1+\sqrt{x}}\right]dx$

$\displaystyle \int \left[\dfrac{1}{\sqrt{x}} - \dfrac{1}{\sqrt{x}(1+\sqrt{x})}\right]dx$
$\downarrow$
$1+\sqrt{x}=t$
$\dfrac{1}{2\sqrt{x}}dx=dt$

$\dfrac{x^{\dfrac{-1}{2}+1}}{-\dfrac{1}{2}+1}-\displaystyle\int\dfrac{2dt}{t}$

$2\sqrt{x}-2\log t$

$2(\sqrt{x}-\log(1+\sqrt{x}))+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 Subjective Medium
Find the integral of the function
${\sin ^2}\left( {2x + 5} \right)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int { { e }^{ x }\left[ f(x)+f'(x) \right] } dx$ is equal to
• A. ${ e }^{ x }+c$
• B. ${ e }^{ x }f'(x)+c$
• C. None of these
• D. ${ e }^{ x }f(x)+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
For any natural number m, $\int { \left( { x }^{ 7m }+{ x }^{ 2m }+{ x }^{ m } \right) { \left( { 2{ x }^{ 6m }+7{ x }^{ m }+14 } \right) }^{ \frac { 1 }{ m } }dx }$ (where x>0), equals
• A. $\displaystyle \frac { { (7{ x }^{ 7m }+2{ x }^{ 2m }+14{ x }^{ m }) }^{ \frac { m+1 }{ m } } }{ 14(m+1) } +C$
• B. $\displaystyle \frac { { (2{ x }^{ 7m }+14{ x }^{ 2m }+7{ x }^{ m }) }^{ \frac { m+1 }{ m } } }{ 14(m+1) } +C$
• C. $\displaystyle \frac { { (7{ x }^{ 7m }+2{ x }^{ 2m }+{ x }^{ m }) }^{ \frac { m+1 }{ m } } }{ 14(m+1) } +C$
• D. $\displaystyle \frac { { (2{ x }^{ 7m }+7{ x }^{ 2m }+14{ x }^{ m }) }^{ \frac { m+1 }{ m } } }{ 14(m+1) } +C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate: $\displaystyle \int e^x (\tan x + \log (\sec x)) dx.$

Evaluate the following : $\displaystyle\int \sqrt{\dfrac{9+x}{9-x}}.dx$