Mathematics

# The value of $\displaystyle \int_{-1}^{1}\dfrac{\cot^{-1}\ x}{\pi}\ dx$ is

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

#### Realted Questions

Q1 Single Correct Medium
$\int { { (cosx) }^{ 4 } } dx=Ax+Bsin2x+Csin4x$ then {A,B,C) equals
• A. $\left\{ \frac { 3 }{ 8 } ,\frac { 1 }{ 2 } ,\frac { 1 }{ 6 } \right\}$
• B. $\left\{ \frac { 3 }{ 4 } ,\frac { 1 }{ 4 } ,\frac { 1 }{ 16 } \right\}$
• C. $\left\{ \frac { 3 }{ 8 } ,\frac { 1 }{ 4 } ,\frac { 1 }{ 32 } \right\}$
• D. $\left\{ \frac { 3 }{ 8 } ,\frac { 1 }{ 4 } ,\frac { 1 }{ 32 } \right\}$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int(\tan x+\log(\sec x)).e^{x}dx=$
• A. $\log(\sec x)+c$
• B. $\tan x.e^{x}+c$
• C. $-e^{x}\log(\sec x)+c$
• D. $e^{x}$. log(Secx)$+$c

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\lim _{ n\rightarrow \infty }{ \dfrac { \left( { 1 }^{ k }+{ 2 }^{ k }+{ 3 }^{ k }+.....+{ n }^{ k } \right) }{ \left( { 1 }^{ 2 }+{ 2 }^{ 2 }+.....+{ n }^{ 2 } \right) \left( { 1 }^{ 3 }+{ 2 }^{ 3 }+.....+{ n }^{ 3 } \right) } }=F(k)$, then $(k\in N)$
• A. $F(5)=0$
• B. $F(6)=\dfrac{12}{7}$
• C. $F(6)=\dfrac{5}{7}$
• D. $F(k)\ is\ finite\ for \ k \le 6$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle\int{\frac{dx}{(x^2+1)(x^2+4)}}=k\tan^{-1}{x}+l\tan^{-1}{\frac{x}{2}}+C$, then
• A. $\displaystyle k=\frac{1}{3}$
• B. $\displaystyle l=\frac{2}{3}$
• C. $\displaystyle l=-\frac{1}{6}$
• D. $\displaystyle k=-\frac{1}{3}$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Medium
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 $\displaystyle f(x)>0 \forall x \in R$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives

1 Verified Answer | Published on 17th 08, 2020