Mathematics

# $\displaystyle\int \dfrac{1}{1+\sin^2 x}dx$

##### SOLUTION
$\int { \cfrac { 1 }{ \sin ^{ 2 }{ x } } } dx=\int { \cfrac { \sec ^{ 2 }{ x } }{ \left( \cfrac { 1 }{ \cos ^{ 2 }{ x } } +\tan ^{ 2 }{ x } \right) } } dx=\int { \cfrac { \sec ^{ 2 }{ x } }{ \sec ^{ 2 }{ x } +\tan ^{ 2 }{ x } } } dx\quad$
put $t=\tan { x } \Rightarrow dt=\sec ^{ 2 }{ x } dx$
$\Rightarrow \int { \cfrac { 1 }{ 1+2{ t }^{ 2 } } } dx=\int { \cfrac { 1 }{ 1+{ \left( \sqrt { 2 } t \right) }^{ 2 } } } dt\Rightarrow \cfrac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \sqrt { 2 } } t\Rightarrow \cfrac { 1 }{ \sqrt { 2 } } \tan ^{ -1 }{ \left( \sqrt { 2 } \tan { x } \right) }$

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

#### Realted Questions

Q1 Single Correct Hard
If c is an arbitrary constant, then$\displaystyle \int \frac{2x}{(1+x^{2})(1+x)^{2}} dx =$

• A. $\displaystyle \tan^{-1} x - \frac{1}{1+x} + c$
• B. $\displaystyle \cot^{-1} x - \frac{1}{1+x} + c$
• C. $\displaystyle \cot^{-1} x + \frac{1}{1+x} + c$
• D. $\displaystyle \tan^{-1} x + \frac{1}{1+x} + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int \dfrac {x^2+1}{x^4+1} dx=$ ?
• A. $\frac {1}{\sqrt{2}}tan^{-1}\left ( x-\frac {1}{x} \right )+C$
• B. $\frac {1}{\sqrt{2}}cot^{-1}\left ( x-\frac {1}{x} \right )+C$
• C. none of these
• D. $\frac {1}{\sqrt{2}}tan^{-1}\left \{ \frac {1}{\sqrt{2}} \left ( x-\frac {1}{x} \right ) \right \} +C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Let $F(x)$ be the primitive of $\displaystyle \frac{3x+2}{\sqrt{x-9}}$ w.r.t $x$ . If $F(10)=60$ then the value of $F(13)$, is
• A. $66$
• B. $248$
• C. $264$
• D. $132$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\int \ sinx\;\; d(\ cos x) =$

Solve $\displaystyle \int\sqrt{\dfrac{a-x}{a+x}}dx$