Mathematics

# $\int { \dfrac { \sqrt { 4+{ x }^{ 2 } } }{ { x }^{ 6 } } dx } =\dfrac { { \left( a+{ x }^{ 2 } \right) }^{ 3/2 }\left( { x }^{ 2 }-b \right) }{ 120{ x }^{ 5 } }+C$ then $a+b$ equals to:

##### SOLUTION
$\displaystyle\int \dfrac{\sqrt{4+x^{2}}}{x^{6}}dx$
Put $x=\dfrac{1}{t}\implies dx=-\dfrac{1}{t^{2}}dt$
$\displaystyle\int \dfrac{\sqrt{4+\dfrac{1}{t^{2}}}}{\dfrac{1}{t^{6}}}\times \dfrac{-dt}{t^{2}}=-\displaystyle \int t^{3}\sqrt{4t^{2}+1}dt$
Put $4{t^{2}}+1=l^{2}\implies t{dt}=\dfrac{l{dl}}{4}$
$\displaystyle\int -\dfrac{l^{2}-1}{4}\times \dfrac{l^{2}dl}{4}=-\dfrac{1}{16}\displaystyle\int (l^{4}-l^{2})dl=-\dfrac{l^{3}(3{l^{2}-5})}{240}=\dfrac{(4+x^{2})^{\dfrac{3}{2}}(x^{2}-\dfrac{3}{2})}{120{x^{5}}}$
$\implies a=4,b=\dfrac{3}{2}\implies a+b=4+\dfrac{3}{2}=\dfrac{11}{2}$

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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 Hard
Evaluate $\displaystyle {\int \sin^{-1}\, \sqrt {\frac {x}{a\, +\, x}} dx}$
• A. $(a + x) \tan^{-1} \displaystyle \sqrt \frac {x}{a}\, -\, \sqrt {ax}\, +\, c$
• B. $a \tan^{-1} \displaystyle \sqrt \frac {x}{a}\, -\, \sqrt {ax}\, +\, c$
• C. $(a + x) \tan^{-1} \displaystyle \sqrt {x}\, -\, \sqrt {ax}\, +\, c$
• D. $\displaystyle \left(\frac { ax+1 }{ 2a } \right)\sin^{ -1 }\, \sqrt { \frac { x }{ a\, +\, x } } -\frac { 1 }{ 2 } \sqrt { \frac { x }{ a } } +c$

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Hard
f(x) = $\displaystyle \int \frac{dx}{x^{2}\left ( 1+x^{5} \right )^{4/5}}$, taking c = 0, value of $|f(1)|=$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate the integral
$\displaystyle \int_{\frac{\sqrt{2}}{3}}^{ \frac{\sqrt{3}}{3}}\displaystyle \frac{dx}{\sqrt{4-9x^{2}}}$
• A. $\displaystyle \frac{\pi}{3}$
• B. $\displaystyle \frac{\pi}{4}$
• C. $\displaystyle \frac{7\pi}{30}$
• D. $\displaystyle \frac{\pi}{36}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate:
$\int { \cfrac { \sec ^{ 2 }{ x } }{ cosec ^{ 2 }{ x } } } dx$

Evaluate $\int \dfrac{e^x-e^{-x}}{e^x+e^{-x}}dx$