Mathematics

$\displaystyle \int_{2}^{4} \dfrac{\sqrt{x^{2}-4}}{x^{4}}dx=$

$\dfrac{\sqrt{3}}{32}$

SOLUTION
$\int_2^4 {\cfrac{{\sqrt {{x^2} - 4} }}{4}}$
$\int_2^4 {\cfrac{{\sqrt {1 - \cfrac{4}{{{x^2}}}dx} }}{{{x^4}}}}$
$\int_2^4 {\cfrac{{\sqrt {1 - \cfrac{4}{{{x^2}}}dx} }}{{{x^3}}}}$
putting $1 - \cfrac{4}{{{x^2}}} = t$
$- 4 \times \cfrac{{ - 2}}{{{x^3}}}dx = dt$
$\cfrac{{dx}}{{{x^3}}} = \cfrac{{dt}}{8}$
$\int_0^{\cfrac{3}{4}} {\cfrac{{\sqrt t }}{8} \times dt}$
$= \cfrac{1}{8} \times \cfrac{2}{3}{\left[ {{t^{\cfrac{3}{2}}}} \right]^\cfrac{3}{4}}$
$= \cfrac{1}{{12}}\left[ {{{\left( {\cfrac{3}{4}} \right)}^\cfrac{3}{2}} - 0} \right]$
$= \cfrac{1}{{12}}\left[ {{{\left( {\cfrac{{\sqrt 3 }}{2}} \right)}^3}} \right]$
$= \cfrac{1}{{12}} \times \cfrac{{3\sqrt 3 }}{8}$
$= \cfrac{{\sqrt 3 }}{{32}}$

Its FREE, you're just one step away

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

Realted Questions

Q1 Subjective Hard
Evaluate  $\int \dfrac {\sin \theta}{\sin 3\theta} d\theta$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\displaystyle \int \dfrac{dx}{(1+x^2)^2}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int \sin ^{-1}(\cos x) d x$ is equal to
• A. $\frac{\pi x}{2}+c$
• B. $\frac{\pi x^{2}}{2}+c$
• C. $\frac{\pi x+x^{2}}{2}+c$
• D. $\frac{\pi x-x^{2}}{2}+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The value of $\displaystyle \sum_{r=1}^{\infty }\tan ^{-1}\frac{2r}{2+r^{2}+r^{4}}$ is equal to
• A. $\displaystyle \frac{\pi }{3}$
• B. $\displaystyle \frac{\pi }{2}$
• C. $\displaystyle \pi$
• D. $\displaystyle \frac{\pi }{4}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Hard
If $\int { f\left( x \right) dx=\Psi \left( x \right) }$, then $\int { { x }^{ 5 }f\left( { x }^{ 3 } \right) } dx$ is equal to:
• A. $\dfrac { 1 }{ 3 } { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -3\int { { x }^{ 3 } } \Psi \left( { x }^{ 3 } \right) dx+C$
• B. $\dfrac { 1 }{ 4 } { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -\int { { x }^{ 2 } } \Psi \left( { x }^{ 3 } \right) dx+C$
• C. $\dfrac { 1 }{ 3 } \left[ { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -\int { { x }^{ 3 } } \Psi \left( { x }^{ 3 } \right) dx \right] +C$
• D. $\dfrac { 1 }{ 3 } { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -\int { { x }^{ 2 } } \Psi \left( { x }^{ 3 } \right) dx +C$