Mathematics

# Evaluate the following integrals : $\int \dfrac{x^{2}}{x^{4}+x^{2}+1}dx$

##### SOLUTION
We have,
$\begin{array}{l} I=\int { \dfrac { { { x^{ 2 } } } }{ { { x^{ 4 } }+{ x^{ 2 } }+1 } } } dx \\ =\int { \dfrac { { dx } }{ { { x^{ 2 } }+1+\dfrac { 1 }{ { { x^{ 2 } } } } } } } \\ =\int { \dfrac { { dx } }{ { { { \left( { { x^{ 2 } }+\dfrac { 1 }{ x } } \right) }^{ 2 } }-1 } } } \\ =\dfrac { 1 }{ 2 } \log \left( { \dfrac { { x+\dfrac { 1 }{ x } -1 } }{ { x+\dfrac { 1 }{ x } +1 } } } \right) \\ =\dfrac { 1 }{ 2 } \log \left( { \dfrac { { { x^{ 2 } }-x+1 } }{ { { x^{ 2 } }+x+1 } } } \right) +C \end{array}$

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

#### Realted Questions

Q1 Subjective Hard
Solve : $\dfrac { \sqrt { x ^ { 2 } + 1 } \left[ \log \left( x ^ { 2 } + 1 \right) - 2 \log x \right] } { x ^ { 4 } }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Solve $\int _{ }^{ }{ { x }^{ 2 }{ e }^{ { x }^{ 3 } } } dx$ equals
• A. $\cfrac { 1 }{ 3 } { e }^{ { x }^{ 2 } }+C$
• B. $\cfrac { 1 }{ 2 } { e }^{ { x }^{ 3 } }+C$
• C. $\cfrac { 1 }{ 2 } { e }^{ { x }^{ 2 } }+C$
• D. $\cfrac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Medium
Let $\displaystyle f(x)=ax^{3}+bx^{2}+cx$ have relative extrema x=1 and at $\displaystyle x=5$.If $\displaystyle \int_{-1}^{1}f(x)dx=6$ then
• A. $c=15$
• B. $a=1$
• C. $a=-1$
• D. $b=9$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The integral $\int_{\pi /4}^{\pi / 2}(2cosec x)^{17}dx$ is equal to
• A. $\int_{0}^{\log(1 + \sqrt {2})} 2(e^{u} + e^{-u})^{17}du$
• B. $\int_{0}^{\log(1 + \sqrt {2})} 2(e^{u} - e^{-u})^{17}du$
• C. $\int_{0}^{\log(1 + \sqrt {2})} 2(e^{u} - e^{-u})^{16}du$
• D. $\int_{0}^{\log(1 + \sqrt {2})} 2(e^{u} + e^{-u})^{16}du$

Let $T=\int_0^{\ln2}\dfrac{2e^{3x}+ 3e^{2x}-6e^x}{6(e^{3x}+e^{2x}-e^x+1)}dx,$ then $e^T=\frac{p}{q}$ where p and q are coprime to each other, then the value of $p+ q$ is