Mathematics

Evaluate the following integral : $$ \displaystyle \int \sqrt{4x^{2}+9} \ dx $$


ANSWER

$$ \dfrac{x}{2}.\sqrt{4x^{2}+9}+\dfrac{9}{4} \ln \left | 2x+\sqrt{4x^{2}+9} \right |+C $$


SOLUTION
Let $$ I=\displaystyle\int \sqrt{4 x^2+9}d x$$

$$\implies I=\displaystyle\int \sqrt{4\left(x^2+\dfrac{9}{4}\right)}d x$$

$$\implies I=2\displaystyle\int \sqrt{x^2+\left(\dfrac{3}{2}\right)^2} d x$$

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 $$a=\dfrac{3}{2}$$

$$\implies I=2\left(\dfrac{x}{2}\sqrt{x^2+\dfrac{9}{4}}+\dfrac{9/4}{2}\ln \left|x+\sqrt{x^2+\dfrac{9}{4}}\right|\right)+C$$

$$\implies I=\dfrac{x}{2}\sqrt{4\bigg(x^2+\dfrac{9}{4}\bigg)}+\dfrac{9}{4}\ln \left|x+\sqrt{\dfrac{4 x^2+9}{4}}\right|+C$$

$$\implies I=\dfrac{x}{2}\sqrt{4 x^2+9}+\dfrac{9}{4}\ln \left|\dfrac{2 x}{2}+\dfrac{\sqrt{ 4 x^2+9}}{2}\right|+C$$

$$\implies I=\dfrac{x}{2}\sqrt{4 x^2+9}+\dfrac{9}{4}\ln \left|2 x+\sqrt{4 x^2+9}\right|+C$$
View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Evaluate: $$\displaystyle \int e^{2x} \sin 3x \ dx $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Evaluate: $$\displaystyle \int { \cfrac { \sec ^{ 2 }{ x }  }{ \tan ^{ 2 }{ x } +4 }  } dx\quad $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
$$\displaystyle \int { \frac { x+\sqrt [ 3 ]{ { x }^{ 2 } } +\sqrt [ 6 ]{ x }  }{ x\left( 1+\sqrt [ 3 ]{ x }  \right)  } dx } $$ is equal to
  • A. $$\displaystyle \frac { 3 }{ 2 } { x }^{ 2/3 }-6\tan ^{ -1 }{ { x }^{ 1/6 } } +c$$
  • B. $$\displaystyle -\frac { 3 }{ 2 } { x }^{ 2/3 }+6\tan ^{ -1 }{ { x }^{ 1/6 } } +c$$
  • C. None of these
  • D. $$\displaystyle \frac { 3 }{ 2 } { x }^{ 2/3 }+6\tan ^{ -1 }{ { x }^{ 1/6 } } +c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
$$\displaystyle\int \frac{x+2}{\sqrt{\left ( 4x-x^{2} \right )}}dx.$$
  • A. $$\displaystyle \sqrt{4x-x^{2}}-4\sin ^{-1}\frac{x-2}{2}.$$
  • B. $$\displaystyle -\sqrt{4x-x^{2}}+2\sin ^{-1}\frac{x-2}{2}.$$
  • C. $$\displaystyle -\sqrt{4x-x^{2}}+4\sin ^{-1}{x-2}.$$
  • D. $$\displaystyle -\sqrt{4x-x^{2}}+4\sin ^{-1}\frac{x-2}{2}.$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
For the next two (02) items that follow :
Consider the integrals $$I_1=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{tan x}}$$ and $$I_2=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{sin x}dx}{\sqrt{sin }x+\sqrt{cos}x}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer