Mathematics

# lf $\displaystyle \int\frac{1}{1+\cot x}dx=_{A}$ Iog $|Sinx+$ Cos $x|+$ Bx $+c$, then $A=\ldots\ldots..,\ B=\ldots\ldots\ldots.$,

$-\displaystyle \frac{1}{2},\frac{1}{2}$

##### SOLUTION
$\displaystyle \int \frac{sin\ x}{sin\ x+cos\ x}dx$
Lets write $N(x)= \lambda (D^{1}(x))+\mu (D(x))$
$\displaystyle sin\ x\ \lambda(cos\ x-sin\ x)+\mu (sin\ x+cos\ x)$
$\displaystyle \lambda +\mu = 0$
$\displaystyle \lambda =-\frac{1}{2} ;\ \mu =\frac{1}{2}$
$\displaystyle -\lambda +\mu =1$
$\displaystyle N(x)=-\frac{1}{2} \ D^{1}(x)+\frac{1}{2} \ D(x)$
$\displaystyle \int \frac{N(x)}{D(x)}\ dx=-\frac{1}{2} \int \frac{D^{1}(x)}{D(X)}+\frac{1}{2} \int \frac{D(X)}{D(x)}\ dx$
$\displaystyle =-\frac{1}{2} \log\ (D(x))+\frac{1}{2} x+c$
$\displaystyle =-\frac{1}{2} \log\ \left | sin\ x+cos\ x \right |+\frac{1}{2} x+c$
$\displaystyle \int \frac{1}{1+cot\ x}\ dx =-\frac{1}{2} log\ \left | sin\ x+cos\ x \right |+\frac{1}{2} x+c$
$\displaystyle A= -\frac{1}{2} ;\ B= \frac{1}{2}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 Single Correct Medium

$\displaystyle \int_{-3\pi/2}^{-\pi/2}[(x+\pi)^{3}+\cos^{2}(x+3\pi)]dx=$
• A. $\displaystyle \frac{\pi}{4}-1$
• B. $\displaystyle \frac{\pi^{4}}{32}$
• C. $\displaystyle \frac{\pi^{4}}{32}+\frac{\pi}{2}$
• D. $\displaystyle \frac{\pi}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle\int \dfrac{\sqrt{x^2-a^2}}{x}dx=$ _______.
• A. $x\sqrt{x^2-a^2}-\dfrac{1}{a}\tan^{-1}\left(\dfrac{x}{a}\right)+c$
• B. $\sqrt{x^2-a^2}+a\sec^{-1}\left(\dfrac{x}{a}\right)+c$
• C. $\sqrt{x^2-a^2}+\dfrac{1}{x}\sec^{-1}(x)+c$
• D. $\sqrt{x^2-a^2}-a\cos^{-1}\left(\dfrac{a}{x}\right)+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate:
$\int { \dfrac { \sin { x-\cos { x } } }{ \sqrt { \sin { 2x } } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\int { \cfrac { dx }{ \left( 16-4{ x }^{ 2 } \right) } } =$?
• A. $\cfrac { 1 }{ 8 } \log { \left| \cfrac { 2-x }{ 2+x } \right| } +C$
• B. $\cfrac { 1 }{ 16 } \log { \left| \cfrac { 2-x }{ 2+x } \right| } +C\quad$
• C. $\cfrac { 1 }{ 8 } \log { \left| \cfrac { 2+x }{ 2-x } \right| } +C$
• D. $\cfrac { 1 }{ 16 } \log { \left| \cfrac { 2+x }{ 2-x } \right| } +C\quad \quad$

$\int (sinx + cos x )^2. dx$