Mathematics

# $\int { \sqrt { 3{ x }^{ 2 }-4x+5 } } dx$

##### SOLUTION

We have,

$\int{\sqrt{3{{x}^{2}}-4x+5}dx}$

$\Rightarrow \int{\sqrt{3\left( {{x}^{2}}-\dfrac{4}{3}x+\dfrac{5}{3} \right)}dx}$

$\Rightarrow \int{\sqrt{3}}\sqrt{\left( {{x}^{2}}-\dfrac{4}{3}x+\dfrac{5}{3} \right)}dx$

On adding and subtracting coefficient of x

Then,

${{\left( \dfrac{4}{6} \right)}^{2}}=\dfrac{4}{9}$

Then,

$\int{\sqrt{3}}\sqrt{{{x}^{2}}-\dfrac{4}{3}x+\dfrac{4}{9}-\dfrac{4}{9}+\dfrac{5}{3}}dx$

$\Rightarrow \int{\sqrt{3}}\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-\dfrac{11}{9}}dx$

$\Rightarrow \int{\sqrt{3}}\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}}dx$

On applying

$\int{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\dfrac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}-\dfrac{{{a}^{2}}}{2}\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|$

$\int{\sqrt{3}}\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}}dx=\dfrac{\left( x-\dfrac{2}{3} \right)}{2}\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}}-\dfrac{{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}}{2}\ln \left| \left( x-\dfrac{2}{3} \right)+\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}} \right|$

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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
$\displaystyle \int(x^{x})^{2}(1+\log x)dx$
• A. $x^{x}+k$
• B. $\displaystyle \frac{x^{x}}{2}+k$
• C. $\displaystyle \frac{x}{2}+c$
• D. $\displaystyle \frac{(x^{x})^{2}}{2}+k$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \frac{1}{\sqrt{\left ( x \right )}\left [ \sqrt{\left ( x \right )}+1 \right ]}dx.$
• A. $\displaystyle 2\log \left ( 1-\sqrt{x} \right ).$
• B. $\displaystyle \log \left ( 1+\sqrt{x} \right ).$
• C. $\displaystyle \sqrt{2}\log \left ( 1+\sqrt{x} \right ).$
• D. $\displaystyle 2\log \left ( 1+\sqrt{x} \right ).$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\displaystyle \int \dfrac{2x+3}{3x+2}dx$ is equal to

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Medium
$\displaystyle \int_{0}^{\pi /2}\frac{a\sin x+b\cos x}{\sin x+\cos x}dx= \left ( a+b \right )\frac{\pi }{C}$
What is C?

$\int \frac{x}{x^2 + a^2} \;dx$