Mathematics

The value of  $$\displaystyle \int_{0}^{\pi /4}\frac{\sec x}{\left ( \sec x+\tan x \right )^{2}}dx$$ is 


ANSWER

none of these


SOLUTION
Let $$\displaystyle I=\int _{ 0 }^{ \dfrac { \pi  }{ 4 }  }{ \frac { \sec { x }  }{ { \left( \sec { x } +\tan { x }  \right)  }^{ 2 } }  } dx$$

Multiply numerator and denominator by $$\cos ^{ 2 }{ x } $$, we get

$$\displaystyle I=\int _{ 0 }^{ \dfrac { 1 }{ \sqrt { 2 }  }  }{ \dfrac { 1 }{ { \left( u+1 \right)  }^{ 2 } }  } du=\left[ \dfrac { -1 }{ u+1 }  \right] _{ 0 }^{ \dfrac { 1 }{ \sqrt { 2 }  }  }$$

$$\displaystyle =-\dfrac { 1 }{ \dfrac { 1 }{ \sqrt { 2 }  } +1 } $$

$$=\dfrac { -\sqrt { 2 }  }{ 1+\sqrt { 2 }  } $$
View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Solve:
$$\int { \dfrac { dx }{ 2{ x }^{ 2 }+x-1 }  } $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
$$ \int \dfrac {x+2}{(x^2 + 3x +3) \sqrt{x+1} } dx $$ is equal to :
  • A. $$ \dfrac {1}{ \sqrt3} \tan^{-1} \left( \dfrac {x} { \sqrt{3(x+1)} } \right) + C $$
  • B. $$ \dfrac {1}{ \sqrt3} \tan^{-1} \left( \dfrac {x} { \sqrt{x+1} } \right) + C $$
  • C. None of these
  • D. $$ \dfrac {2}{ \sqrt3} \tan^{-1} \left( \dfrac {x} { \sqrt{3(x+1)} } \right) + C $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 One Word Hard
The value of $$\displaystyle \frac{\left ( \sqrt{2}+1 \right )198}{\pi }\int_{\pi /4}^{3\pi /4}\displaystyle \frac{\phi }{1 + \sin \phi }\:d\phi $$ is

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Hard
Evaluate:
$$\int { \cfrac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
$$\int_\limits{\frac{\pi}{2}}^{\pi}\dfrac{1-\sin x}{1- \cos x}dx$$
  • A. $$-\log(2)-1$$
  • B. $$\dfrac{1}{\log(2)}+1$$
  • C. $$\dfrac{1}{\log(2)}-1$$
  • D. $$-\log(2)+1$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer