Mathematics

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

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

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

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

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

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

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

1 Verified Answer | Published on 17th 09, 2020

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. $\dfrac {2}{ \sqrt3} \tan^{-1} \left( \dfrac {x} { \sqrt{3(x+1)} } \right) + C$

1 Verified Answer | Published on 17th 09, 2020

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

1 Verified Answer | Published on 17th 09, 2020

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

1 Verified Answer | Published on 17th 09, 2020

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. $-\log(2)+1$