Mathematics

# $\int { x+2y } \sqrt { { x }^{ 2 }+5x+6 }$

##### SOLUTION
$\displaystyle \int x+2y \sqrt {x^2 +5x+6}dx$
$\Rightarrow \ \displaystyle \int x\ dx +2y \displaystyle \int \sqrt {x^2+5x+6}dx$
$\Rightarrow \ \dfrac {x^2}{2}+2y \displaystyle \int \sqrt {\left (x+\dfrac {5}{2}\right)^2 -\dfrac {1}{4}}dx\quad \mu =x+\dfrac {5}{2}$
$\Rightarrow \ 2y \dfrac {1}{2} \displaystyle \int \sqrt {4\mu^2 -1}du \quad v= \sec^{-1}(2\mu)$
$\Rightarrow \ 2y.\dfrac {1}{2}.\dfrac {1}{2} \displaystyle \int (-1+\sec^2 (v))\sec v\ dv$
$\displaystyle \int \sec v \ dv +\displaystyle \int \sec^3 x\ dv$
$\dfrac {\sec^2 v \sin v}{2}+\dfrac {1}{2} \displaystyle \int \sec v\ dv$
$\Rightarrow \ \dfrac {x^2}{2}\dfrac {y}{2} (-\ln |\tan v+ \sec v|+ \dfrac {\sec^2 v.\sin v}{2}+\dfrac {1}{2} \ln |\tan v+\sec v|)$

$\Rightarrow \dfrac { 1 }{ 2 } \left[ { x }^{ 2 }+y\left( \ln { \left| \tan { \left( \sec ^{ -1 }{ \left( 2\left( x+\dfrac { 5 }{ 2 } \right) \right) } \right) } \right| } \right) +\sec { \left( \sec ^{ -1 }{ \left( 2\left( x+\dfrac { 5 }{ 2 } \right) \right) } \right) } \right] \left( -1+\dfrac { 1 }{ 2 } \right) +\sec ^{ 2 }{ \left( \sec ^{ -1 }{ \left( 2\left( x+\dfrac { 5 }{ 2 } \right) \right) } \right) } \sin { \left( \sec ^{ -1 }{ 2\left( x+\dfrac { 5 }{ 2 } \right) } \right) } \\$
$\Rightarrow \dfrac { { x }^{ 2 } }{ 2 } +\dfrac { y }{ 4 } \left[ 2\left( 2x+5 \right) \sqrt { { x }^{ 2 }+5x+6 } -\ln { \left| 2\sqrt { { x }^{ 2 }+5x+6 } +2x+5 \right| } \right] +C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Hard
$\int { \cfrac { { x }^{ 2 } }{ \sqrt { { x }^{ 6 }-1 } } dx } =$?
• A. $\cfrac { 1 }{ 2 } \log { \left| { x }^{ 3 }+\sqrt { { x }^{ 6 }-1 } \right| } +C$
• B. $\cfrac { 1 }{ 3 } \log { \left| { x }^{ 3 }-\sqrt { { x }^{ 6 }-1 } \right| } +C$
• C. none of these
• D. $\cfrac { 1 }{ 3 } \log { \left| { x }^{ 3 }+\sqrt { { x }^{ 6 }-1 } \right| } +C$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Evaluate : $\displaystyle \int \sqrt{\dfrac{a - x}{x}} dx$
• A. $\dfrac{\sqrt{a-x}}{\sqrt{x}}+a\tan^{-1}\left(\dfrac{\sqrt{x}}{\sqrt{a-x}}\right)+c$
• B. $\dfrac{\sqrt{a-x}}{\sqrt{x}}-a\tan^{-1}\left(\dfrac{\sqrt{x}}{\sqrt{a-x}}\right)+c$
• C. $\sqrt{a-x} \sqrt{x}-a\tan^{-1}\left(\dfrac{\sqrt{x}}{\sqrt{a-x}}\right)+c$
• D. $\sqrt{a-x} \sqrt{x}+a\tan^{-1}\left(\dfrac{\sqrt{x}}{\sqrt{a-x}}\right)+c$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int {\left( {\frac{1}{{\left( {\ell nx} \right)}} - \frac{1}{{{{(\ell nx)}^2}}}} \right)\,dx}$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium

$\displaystyle \overset{\infty}{\underset{0}{\int}} x^6 e^{\tfrac{-x}{2}} dx =$

• A. $2^6 \left \lfloor 6 \right.$
• B. $\dfrac{ \left \lfloor 6 \right.}{2^7}$
• C. $\dfrac{ \left \lfloor 6 \right.}{2^6}$
• D. $2^7 \left \lfloor 6 \right.$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Subjective Medium
$\int x ^ { 5 } d x$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020