Mathematics

# Evaluate the following integrals:$\int { \sqrt { 2{ x }^{ 2 }+3x+4 } } dx$

##### SOLUTION
Let $I=\displaystyle\int \sqrt{2 x^2+3 x+4} d x$

$=\displaystyle\int \sqrt{2\left(x^2+2(x)\bigg(\dfrac{3}{4}\bigg)+2\right)}$

$=\sqrt{2}\displaystyle\int \sqrt{x^2+2(x)\bigg(\dfrac{3}{4}\bigg)+\bigg(\dfrac{3}{4}\bigg)^2-\bigg(\dfrac{3}{4}\bigg)^2+2}\ d x$

$=\sqrt{2}\displaystyle\int \sqrt{\bigg(x+\dfrac{3}{4}\bigg)^2+\dfrac{23}{16}}\ d x$                                                        $(\because (a+b)^2=a^2+2 a b+b^2)$

Put $t=x+\dfrac{3}{4}\implies d t=d x$

$I=\sqrt{2}\displaystyle\int \sqrt{t^2+\bigg(\dfrac{\sqrt{23}}{4}\bigg)^2}\ d t$

As we know that

$\displaystyle\int \sqrt{a^2+x^2} d x=\dfrac{x}{2}\sqrt{x^2+a^2}+\dfrac{a^2}{2}\ln |x+\sqrt{x^2+a^2}|+C$

Here $a=\dfrac{\sqrt{23}}{4}$

$\implies I=\sqrt{2}\left(\dfrac{t}{2}\sqrt{t^2+\dfrac{23}{16}}+\dfrac{23/16}{2}\ln \left|t+\sqrt{t^2+\dfrac{23}{16}}\right|+C\right)$

$\implies I=\dfrac{1}{2}\bigg(x+\dfrac{3}{4}\bigg)\sqrt{2\bigg(x+\dfrac{3}{4}\bigg)^2+\dfrac{23}{16}}+\dfrac{23\sqrt{2}}{32}\ln \left|x+\dfrac{3}{4}+\sqrt{\bigg(x+\dfrac{3}{4}\bigg)^2+\dfrac{23}{16}}\ \right|+C$

$\implies I=\dfrac{1}{8}(4 x+3)\sqrt{2 x^2+3 x+4}+\dfrac{23\sqrt{2}}{32}\ln \left|x+\dfrac{3}{4}+\sqrt{x^2+\dfrac{3}{2} x+2}\ \right|+C$

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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
If $\displaystyle I_{1}=\int_{1}^{2}x\left(\sqrt{x}+\sqrt{3-x}\right)dx$ and $I_{2}=\displaystyle \int_{1}^{2}\left(\sqrt{x}+\sqrt{3-x}\right)dx$, then $\displaystyle \frac{I_{1}}{I_{2}} =$
• A. $\dfrac12$
• B. $2$
• C. $1$
• D. $\dfrac32$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\displaystyle \frac { d }{ dx } \left( \phi \left( x \right) \right) =f\left( x \right)$, then $\displaystyle \int _{ 1 }^{ 2 }{ f\left( x \right) }$ is equal to:
• A. $\displaystyle f\left( 1 \right) -f\left( 2 \right)$
• B. $\displaystyle \phi \left( 1 \right) -\phi \left( 2 \right)$
• C. $\displaystyle f\left( 2 \right) -f\left( 1 \right)$
• D. $\displaystyle \phi \left( 2 \right) -\phi \left( 1 \right)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate the integral
$\displaystyle \int_{{\pi }/{6}}^{{\pi }/{3}}\frac{dx}{1+\tan ^{11}x}$
• A. $\displaystyle \frac{\pi }{3}$
• B. $1$
• C. None of the above
• D. $\displaystyle \frac{\pi }{12}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\int { \left( x+2 \right) \sqrt { 3x+5 } } dx$

$\displaystyle\int \left(e^x\right)^2 e^x dx$ is equal to