Mathematics

# If $\displaystyle \int \dfrac{x + \sqrt[3]{x^2} + \sqrt[6]{x}}{x(1+\sqrt[3]{x})} dx = ax^{\frac{2}{3}} + b\ \tan^{-1}(\sqrt[6]{x}) +c$ then:

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

##### SOLUTION
$\int { \dfrac { x+{ x }^{ 2/3 }+{ x }^{ 1/6 } }{ x\left( { x }^{ 1/3 }+1 \right) } } dx$
$u={ x }^{ 1/6 }$
$du=\dfrac { 1 }{ 6 } \dfrac { dx }{ { x }^{ 5/6 } }$
${ x }^{ 1/3 }={ u }^{ 2 }$
${ x }^{ 2/3 }={ u }^{ 4 }$
$x={ u }^{ 6 }$
$x={ u }^{ 6 }$
$=6\int { \dfrac { { u }^{ 5 }+{ u }^{ 3 }+1 }{ { u }^{ 2 }+1 } } du$
Polynomial division
$\Rightarrow 6\int { \left( \dfrac { 1 }{ { u }^{ 2 }+1 } +{ u }^{ 3 } \right) } du$
$\Rightarrow 6{ \tan }^{ -1 }\left( u \right) +\dfrac { { 6u }^{ 4 } }{ 4 } +c$
$=6{ \tan }^{ -1 }\left( { x }^{ 1/6 } \right) +\dfrac { { 3x }^{ 2/3 } }{ 2 } +c$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Single Correct Medium
I : Number of partial fractions of $\displaystyle \frac{x^{3}+x^{2}+1}{x^{4}+x^{2}+1}$ is 4
II : Number of partial fractions of $\displaystyle \frac{3x+5}{(x-1)^{2}(x^{2}+1)^{3}}$ is 5

Which of the above statement is true.
• A. only I
• B. Both I and II
• C. Neither I nor II
• D. Only II

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle\int e^{\sin x}\cdot\left(\begin{matrix} \dfrac{sin x+1}{sec x}\end{matrix}\right)dx$ is equal to?
• A. $\cos x\cdot e^{\sin x}+c$
• B. $e^{\sin x}+c$
• C. $e^{\sin x}(\sin x+1)+c$
• D. $\sin x\cdot e^{\sin x}+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Solve $\displaystyle\int {\dfrac{{{x^2}}}{{{{\left( {a + bx} \right)}^2}}}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle \int \dfrac{x + \sqrt[3]{x^2} + \sqrt[6]{x}}{x(1+\sqrt[3]{x})} dx = ax^{\frac{2}{3}} + b\ \tan^{-1}(\sqrt[6]{x}) +c$ then:
• A. $a= \frac{3}{4}$
• B. $a= \frac{-3}{2}$
• C. $a= \frac{-3}{4}$
• D. $a= \frac{3}{2}$

Solve$\displaystyle \int_0^1 \dfrac{x^2-2}{x^2+1}dx$