Mathematics

# The integral $\int \dfrac {2x^{12} + 5x^{9}}{(x^{5} + x^{3} + 1)^{3}} dx$ is equal to:

$\dfrac {x^{10}}{2(x^{5} + x^{3} + 1)^{2}} + C$

##### SOLUTION
$\int \dfrac {2x^{12} + 5x^{9}}{(x^{5} + x^{3} + 1)^{3}} dx$
$=\int \dfrac {2x^{12} + 5x^{9}}{x^{15}(x^{-5} + x^{-2} + 1)^{3}} dx$
$=\int \dfrac {2x^{-3} + 5x^{-6}}{(x^{-5} + x^{-2} + 1)^{3}} dx$

Let $x^{-5}+x^{-2}+1 =t$

$\Rightarrow \int \dfrac{-dt}{t^3}= \dfrac{1}{2t^2}+c$

$=\dfrac {x^{10}}{2(x^{5} + x^{3} + 1)^{2}} +c$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle \int \sqrt{e^{x}-1}\:dx$ is equal to
• A. $\displaystyle \sqrt{e^{x}-1}-\tan^{-1}\sqrt{e^{x}-1}+C$
• B. $\displaystyle \sqrt{e^{x}-1}+\tan^{-1}\sqrt{e^{x}-1}+C$
• C. $\displaystyle 2\left [ \sqrt{e^{x}-1}+\tan^{-1}\sqrt{e^{x}-1} \right ]+C$
• D. $\displaystyle 2\left [ \sqrt{e^{x}-1}-\tan^{-1}\sqrt{e^{x}-1} \right ]+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Hard
The integral $\int_{0}^{1}tan^{-1}\left ( \displaystyle \frac{2x-1}{1+x-x^{2}} \right )dx$ simlifies to a value:

1 Verified Answer | Published on 17th 09, 2020

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. $\displaystyle \frac { 3 }{ 2 } { x }^{ 2/3 }+6\tan ^{ -1 }{ { x }^{ 1/6 } } +c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The value of $\displaystyle \int_{-2}^{2}\left[p\:log\left(\frac{1-x}{1+x}\right)^{-1}+q\:log\left(\frac{1-x}{1+x}\right)^{2}+r\right]dx$ depends on
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Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$