Mathematics

# $\displaystyle \int _{ 0 }^{ x }{ \cfrac { \sin { x } }{ 1+\cos ^{ 2 }{ x } } } dx=\pi \cfrac { \cos { \alpha } }{ 1-\sin ^{ 2 }{ \alpha } }$

for exactly one $\alpha$ in $\left( 0,\pi /2 \right)$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Subjective Medium
Evaluate the following integral:

$\displaystyle\int_{0}^{2}x\sqrt{x+2}dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Evaluate the following integrals:$\displaystyle \int \sqrt{3x^{2}+4}dx$
• A. $\dfrac{x}{2}.\sqrt{3x+4}+\dfrac{2}{\sqrt{3}} \log\left | \sqrt{3}x+\sqrt{3x^{2}+4} \right |+C$
• B. $\dfrac{x}{2}.\sqrt{3x^{2}+4}+\dfrac{4}{\sqrt{3}} \log\left | \sqrt{3}x+\sqrt{3x^{2}+4} \right |+C$
• C. None of these
• D. $\dfrac{x}{2}.\sqrt{3x^{2}+4}+\dfrac{2}{\sqrt{3}} \log\left | \sqrt{3}x+\sqrt{3x^{2}+4} \right |+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following definite integral :
$\int_0 ^{\pi/2}$  $\dfrac {cos ^{5}x} {sin ^{5} x + cos ^{5} x} dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If $I_n=\displaystyle\int^1_0\dfrac{dx}{(1+x^2)^n}; n\in N$, then which of the following statements hold good?
• A. $I_2=\dfrac{\pi}{8}+\dfrac{1}{4}$
• B. $I_2=\dfrac{\pi}{8}-\dfrac{1}{4}$
• C. $I_3=\dfrac{\pi}{16}-\dfrac{5}{48}$
• D. $2n I_{n+1}=2^{-n}+(2n-1)I_n$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int {\cfrac{{dx}}{{{x^2}{{\left( {1 + {x^5}} \right)}^{4/5}}}}}$ is equal to:
• A. $- \cfrac{{{{\left( {1 + {x^5}} \right)}^{1/5}}}}{{5x}} + {\rm{C}}$
• B. $\cfrac{{{{\left( {1 + {x^5}} \right)}^{1/5}}}}{{5x}} + {\rm{C}}$
• C. $\cfrac{{{{\left( {1 + {x^5}} \right)}^{1/5}}}}{x} + {\rm{C}}$
• D. $- \cfrac{{{{\left( {1 + {x^5}} \right)}^{1/5}}}}{x} + {\rm{C}}$