Mathematics

# If $\displaystyle I=\int { \left( \sqrt { \tan { x } } +\sqrt { \cot { x } } \right) } dx=f\left( x \right)+c$

$\sqrt { 2 } \sin ^{ -1 }\times \left( \sin { x } -\cos { x } \right)$

##### SOLUTION
Given,

$\displaystyle \int \sqrt{\tan x}+\sqrt{\cot x}dx$

$\displaystyle \int \dfrac{\sin x+\cos x}{\sqrt{\sin x\cos x}}dx$

put $\sin x-\cos x=u\rightarrow du=\sin x+\cos x dx$

$u^2=\sin^2 x+\cos^2 x-2\sin x\cos x\Rightarrow \sin x\cos x=\dfrac {1-u^2}{2}$

$\displaystyle I=\int \dfrac{\sqrt{2}du}{\sqrt{1-u^2}}$

$=\sqrt{2}\sin ^{-1}u+c$

$=\sqrt{2}\sin^{-1}(\sin x-\cos x)+c$

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

#### Realted Questions

Q1 Single Correct Hard
The value of $\displaystyle\int { \cfrac { \cos { \sqrt { x } } }{ \sqrt { x } } } dx$ is
• A. $2\cos { \sqrt { x } } +C$
• B. $\sqrt { \cfrac { \cos { x } }{ x } } +C$
• C. $\sin { \sqrt { x } } +C$
• D. $2\sin { \sqrt { x } } +C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle\int { \cfrac { 1 }{ 7 } \sin { \left( \cfrac { x }{ 7 } +10 \right) } dx }$ is equal to
• A. $\cfrac { 1 }{ 7 } \cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$
• B. $-\cfrac { 1 }{ 7 } \cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$
• C. $-7\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$
• D. $\cos { \left( x+70 \right) } +C$
• E. $-\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int\frac{1}{\sin x\sqrt{\sin x\cos x}}dx=$
• A. $\sqrt{\tan x}+c$
• B. $-2\sqrt{\tan x}+c$
• C. $2\sqrt{\cot x}+c$
• D. $-2\sqrt{\cot x}+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If $I=\int _{ 2 }^{ 3 }{ \dfrac { { 2x }^{ 5 }+{ x }^{ 4 }-{ 2x }^{ 3 }+{ 2x }^{ 2 }+1 }{ \left( { x }^{ 2 }+1 \right) \left( { x }^{ 4 }-1 \right) } } dx$, then 1 equals
• A. $\dfrac { 1 }{ 2 } ln6-\dfrac { 1 }{ 10 }$
• B. $\dfrac { 1 }{ 2 } ln3-\dfrac { 1 }{ 10 }$
• C. $\dfrac { 1 }{ 2 } ln2+\dfrac { 1 }{ 10 }$
• D. $\dfrac { 1 }{ 2 } ln6+\dfrac { 1 }{ 10 }$

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