Mathematics

# $\int { \dfrac { 1 }{ \sqrt { 3x+5 } -\sqrt { 3x+2 } } } dx =$

##### SOLUTION

We have,

$\int{\dfrac{1}{\sqrt{3x+5}-\sqrt{3x+2}}dx}$

On rationalize and we get,

$\int{\dfrac{1}{\sqrt{3x+5}-\sqrt{3x+2}}\times \dfrac{\sqrt{3x+5}+\sqrt{3x+2}}{\sqrt{3x+5}+\sqrt{3x+2}}dx}$

$=\int{\dfrac{\sqrt{3x+5}+\sqrt{3x+2}}{{{\left( \sqrt{3x+5} \right)}^{2}}-{{\left( \sqrt{3x+2} \right)}^{2}}}}dx$

$=\int{\dfrac{\sqrt{3x+5}+\sqrt{3x+2}}{3x+5-3x-2}}dx$

$=\dfrac{1}{3}\int{\sqrt{3x+5}dx+\dfrac{1}{3}\int{\sqrt{3x+2}dx}}$

$=\dfrac{1}{3}{{\int{\left( 3x+5 \right)}}^{\dfrac{1}{2}}}dx+\dfrac{1}{3}{{\int{\left( 3x+2 \right)}}^{\dfrac{1}{2}}}dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\therefore \int{{{\left( ax+b \right)}^{n}}=}\dfrac{{{\left( ax+b \right)}^{n+1}}}{a\left( n+1 \right)}$

$=\dfrac{1}{3}\dfrac{{{\left( 3x+5 \right)}^{\dfrac{1}{2}+1}}}{3\left( \dfrac{1}{2}+1 \right)}+\dfrac{1}{3}\dfrac{{{\left( 3x+2 \right)}^{\dfrac{1}{2}+1}}}{3\left( \dfrac{1}{2}+1 \right)}$

$=\dfrac{1}{9}\dfrac{{{\left( 3x+5 \right)}^{\dfrac{3}{2}}}}{\dfrac{3}{2}}+\dfrac{1}{9}\dfrac{{{\left( 3x+2 \right)}^{\dfrac{3}{2}}}}{\dfrac{3}{2}}$

$=\dfrac{2}{9}\left[ {{\left( 3x+5 \right)}^{\dfrac{3}{2}}}+{{\left( 3x+2 \right)}^{\dfrac{3}{2}}} \right]$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
$\lim _{ n\rightarrow \infty }{ \cfrac { 1 }{ \sqrt { n } \sqrt { n+1 } } +\cfrac { 1 }{ \sqrt { n } \sqrt { n+2 } } +....+\cfrac { 1 }{ \sqrt { n } \sqrt { 4n } } }$ is equal to
• A. $2$
• B. $4$
• C. $\sqrt { 2 } -1$
• D. $2\left( \sqrt { 5 } -1 \right)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Integrate $\displaystyle \int { \dfrac { { \sec }^{ 2 }x }{ \tan x } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
Primitive of $\displaystyle \sqrt[3]{\frac{x}{(x^{4} - 1)^{4}}}$  w.r.t  is
• A. $\displaystyle \frac{-3}{4}(1 + x^{-4})^{\tfrac{1}{3}} + C$
• B. $\displaystyle \frac{-3}{4}(1 + x^{-4})^{\tfrac{-1}{3}} + C$
• C. $\displaystyle \frac{3}{4}(1 - x^{-4})^{\tfrac{1}{3}} + C$
• D. $\displaystyle \frac{-3}{4}(1 - x^{-4})^{\tfrac{-1}{3}} + C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate the following integrals:
$\displaystyle \int { \cfrac { \cos { x } }{ \sqrt { 4+\sin ^{ 2 }{ x } } } } dx$

$\int{\cos ^4}2x$