Mathematics

# $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sin ^{ \frac { 3 }{ 2 } }{ x } }{ \sin ^{ \frac { 3 }{ 2 } }{ x } +\cos ^{ \frac { 3 }{ 2 } }{ x } } } dx=\dfrac{\pi}{4}\\$

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

#### Realted Questions

Q1 Multiple Correct Hard
$\int { { 5 }^{ mx }{ 7 }^{ nx } } dx,m,n\in N$ is equal to
• A. $\cfrac { \left( m.n \right) { 5 }^{ mx }+{ 7 }^{ nx } }{ m\log { 5 } +n\log { 7 } } +K\quad$
• B. None of these
• C. $\cfrac { { 5 }^{ mx }+{ 7 }^{ nx } }{ m\log { 5 } +n\log { 7 } } +K$
• D. $\cfrac { { e }^{ \left( m\log { 5 } +n\log { 7 } \right) x } }{ \log { { 5 }^{ m } } +\log { { 7 }^{ n } } } +K$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Let $f$ be a positive function. If $\displaystyle I_{1}=\int_{1-k}^{k}xf{x(1-x)}dx,\ I_{2}=\int_{1-k}^{k}f{x(1-x)}dx,$ where $2k-1>0,$ then $\displaystyle \frac{I_{1}}{I_{2}}$ is
• A. $2$
• B. $k$
• C. $1$
• D. $\displaystyle \frac{1}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate  :
$\int {\frac{{x - 5}}{{\sqrt {{x^2} + 6x + 7} }}dx} .$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle \int \dfrac {1}{x^{2}(x^{4} + 1)^{3/4}} dx$ is equal to ____
• A. $\dfrac {-(1 + x^{4})^{1/4}}{x^{2}} + C$
• B. $\dfrac {-(1 + x^{4})^{1/4}}{2x} + C$
• C. $\dfrac {-(1 + x^{4})^{3/4}}{x} + C$
• D. $\dfrac {-(1 + x^{4})^{1/4}}{x} + C$

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