Mathematics

# $\int \dfrac{x^2 +1}{x^4 + x^2 +1} dx$ is equal to

$tan^{-1} ( \dfrac{x^2-1}{\sqrt{3}x}) + c$

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

#### Realted Questions

Q1 Single Correct Medium
The solution for x of the equation $\displaystyle \int_{\sqrt{2}}^{x}\frac{dt}{t\sqrt{t^{2}-1}}=\frac{\pi }{2}$ is
• A. $\displaystyle \frac{\sqrt{3}}{2}$
• B. $2$
• C. $\pi$
• D. $-\sqrt{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Using integral $\int _{ 0 }^{ \pi /2 }{ \ln { \left( \sin { x } \right) } } dx=\int _{ 0 }^{ \pi /2 }{ \ln { \left( \sec { x } \right) } } dx=-\cfrac { \pi }{ 2 } \ln { 2 }$
Evaluate $\int _{ -\pi /4 }^{ \pi /4 }{ \ln { \left( \cfrac { \sin { x } +\cos { x } }{ \cos { x } -\sin { x } } \right) } dx= }$
• A. $\pi \ln{2}$
• B. $\cfrac{\pi \ln{2}}{2}$
• C. $-\pi \ln{2}$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
Primitive of $\displaystyle \dfrac{3x^4-1}{(x^4+x+1)^2}$ w.r.t  $x$ is
• A. $\displaystyle \dfrac{x}{(x^4+x+1)}+c$
• B. $\displaystyle \dfrac{x+1}{(x^4+x+1)}+c$
• C. $-\displaystyle \dfrac{x+1}{(x^4+x+1)}+c$
• D. $-\displaystyle \dfrac{x}{(x^4+x+1)}+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The value of $\displaystyle \int_{-\pi /2}^{\pi /2}\log \left ( \frac{2-\sin \theta }{2+\sin \theta } \right )d\theta$is
• A. $1$
• B. $2$
• C. None of these
• D. $0$

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 $\displaystyle f(x)>0 \forall x \in R$