Mathematics

# Evaluate the following integral:$\int { \left( \sqrt { x } \left( a{ x }^{ 2 }+bx+c \right) \right) } dx$

##### SOLUTION
Given $\displaystyle\int{\left(\sqrt{x}\left(a{x}^{2}+bx+c\right)\right)dx}$

$=\displaystyle\int{\left(a{x}^{\frac{1}{2}+2}+b{x}^{1+\frac{1}{2}}+c{x}^{\frac{1}{2}}\right)dx}$

$=\displaystyle\int{\left(a{x}^{\frac{1+4}{2}}+b{x}^{\frac{2+1}{2}}+c{x}^{\frac{1}{2}}\right)dx}$

$=\displaystyle\int{\left(a{x}^{\frac{5}{2}}+b{x}^{\frac{3}{2}}+c{x}^{\frac{1}{2}}\right)dx}$

$=a\displaystyle\int{{x}^{\frac{5}{2}}dx}+b\displaystyle\int{{x}^{\frac{3}{2}}dx}+c\displaystyle\int{{x}^{\frac{1}{2}}dx}$

We know that $\displaystyle\int{{x}^{n}dx}=\dfrac{{x}^{n+1}}{n+1}+c$

$=\dfrac{a{x}^{\frac{5}{2}+1}}{\dfrac{5}{2}+1}+\dfrac{b{x}^{\frac{3}{2}+1}}{\dfrac{3}{2}+1}+\dfrac{c{x}^{\frac{1}{2}+1}}{\dfrac{1}{2}+1}+C$ where $C$ is the constant of integration.

$=\dfrac{a{x}^{\frac{5+2}{2}}}{\dfrac{5+2}{2}}+\dfrac{b{x}^{\frac{3+2}{2}}}{\dfrac{3+2}{2}}+\dfrac{c{x}^{\frac{1+2}{2}}}{\dfrac{1+2}{2}}+C$

$=\dfrac{a{x}^{\frac{7}{2}}}{\dfrac{7}{2}}+\dfrac{b{x}^{\frac{5}{2}}}{\dfrac{5}{2}}+\dfrac{c{x}^{\frac{3}{2}}}{\dfrac{3}{2}}+C$

$=\dfrac{2a{x}^{\frac{7}{2}}}{7}+\dfrac{2b{x}^{\frac{5}{2}}}{5}+\dfrac{2c{x}^{\frac{3}{2}}}{3}+C$

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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 Medium
$\displaystyle \int{\dfrac{1}{\cos^{2}x(1-\tan x)^{2}}dx}$=
• A. $\dfrac{1}{1-\tan x}+c$
• B. $-\dfrac{1}{3}\dfrac{1}{(1-\tan x)^{3}}+c$
• C. $none\ of\ these$
• D. $\dfrac{1}{\tan x-1}+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int\frac{x-\sin x}{1-\cos x}dx=$
• A. $log |1-Cosx | +c$
• B. $log | x - sin x | +c$
• C. $x\displaystyle \tan\frac{x}{2}+c$
• D. $-x\displaystyle \cot\frac{x}{2}+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following integral
$\int { \cfrac { \sin { \left( x+a \right) } }{ \sin { \left( x+b \right) } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Evaluate: $\displaystyle \int { \dfrac { x\sqrt { x } .dx }{ \sqrt { 1-{ x }^{ 5 } } } }$
• A. $\dfrac { 2 }{ 5 } x\sin ^{ -1 } ({ x } ^{ \frac { 5 }{ 2 } })+c$
• B. $\dfrac { 1 }{ 5 } x\sin ^{ -1 }({ x } ^{ \frac { 3 }{ 2 } })+c$
• C. $\dfrac { 1 }{ 3 } x\sin ^{ -1 }({ x } ^{ \frac { 3 }{ 2 } })+c$
• D. $\dfrac { -2 }{ 5 }\sin ^{ -1 }(\sqrt{1-{ x }^5 })+c$

1 Verified Answer | Published on 17th 09, 2020

Q5 Subjective Medium
Evaluate: $\displaystyle\int \log \left( {\log x} \right) + {\left( {\log x} \right)^{ - 2}}= ?$