Mathematics

# $\int { \dfrac { \sin { 2x } }{ \left( 1+\sin { x } \right) \left( 2+\sin { x } \right) } } dx$

##### SOLUTION

Let $I=\int { \dfrac { sin2x }{ \left( 1+sinx \right) \left( 2+sinx \right) } } dx$

$I=\int { \dfrac { sin2xcosx }{ \left( 1+sinx \right) \left( 2+sinx \right) } } dx$

put $sinx=t$

$cosxdx=dt$

$I=\int { \dfrac { 2tdt }{ \left( 1+t \right) \left( 2+t \right) } }$

$\int { \dfrac { 2tdt }{ \left( 1+t \right) \left( 2+t \right) } } =\dfrac { A }{ \left( 1+t \right) } +\dfrac { B }{ \left( 2+t \right) }$

$2t=A\left( 2+t \right) +B\left( 1+t \right)$

$2t=2A+At+B+Bt$

$2t=\left( A+B \right) t+2A+B$

comparing the coefficients,

$A+B=2\Longrightarrow (1)$

comparing constants,

$2A+B=0\Longrightarrow (2)$

Subtracting (1) from (2)

$A=-2$

$B=4$

$\dfrac { 2t }{ \left( 1+t \right) \left( 2+t \right) } =\dfrac { -2 }{ \left( 1+t \right) } +\dfrac { 4 }{ \left( 2+t \right) }$

By integrating,

$\int { \dfrac { 2t }{ \left( 1+t \right) \left( 2+t \right) } } dt=-2\int { \dfrac { dt }{ \left( 1+t \right) } +4\int { \dfrac { dt }{ \left( 2+t \right) } } } =-2log\left| 1+t \right| +4log\left| 2+t \right| +C$

$I=-2log\left| 1+sinx \right| +4log\left| 2+t \right| +C$

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Subjective Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Medium
If $f(x)={ \{ }_{ 0, \quad \quad \quad |x|>1\quad \quad \quad \quad \quad \quad \quad \quad \quad }^{ 1-|x|,\quad \quad |x|<1 }$ and $g(x)=f(x-1)+f(x+1)$ then $\int _{ 0 }^{ 3 }{ g(x)dx }$ is equal to
• A. 1
• B. 2
• C. 3
• D.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Solve:
$\int_{\dfrac{\pi}{2}}^ {\dfrac{\pi}{2}}tanx^{3}\ dx=$
• A. $1$
• B. $\dfrac{1}{2}$
• C. $2$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
If $I_{1}=\displaystyle \int_{e}^{e^{2}}{\dfrac{dx}{\ln x}}$ and $I_{2}=\displaystyle \int_{1}^{2}{\dfrac{e^{x}}{x}dx}$ , then
• A. $2I_{1}=I_{2}$
• B. $I_{1}=2I_{2}$
• C. $2I_{1}=3I_{2}$
• D. $I_{1}=I_{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium

$\displaystyle \int_{0}^{\pi}e^{x}\sin xdx=$
• A. $\displaystyle \frac{1}{2}e^{\pi}$
• B. $e^{\pi}+1$
• C. $\displaystyle \frac{1}{2}(e^{\pi}-1)$
• D. $\displaystyle \frac{1}{2}(e^{\pi}+1)$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Hard
$\int { \cfrac { \cos { x } }{ \sqrt { \sin ^{ 2 }{ x } -2\sin { x } -3 } } } dx=$?
• A. $\log { \left| \sin { x } +\sqrt { \sin ^{ 2 }{ x } -2\sin { x } -3 } \right| } +C$
• B. $\log { \left| \left( \sin { x } -1 \right) -\sqrt { \sin ^{ 2 }{ x } -2\sin { x } -3 } \right| } +C\quad$
• C. none of these
• D. $\log { \left| \left( \sin { x } -1 \right) +\sqrt { \sin ^{ 2 }{ x } -2\sin { x } -3 } \right| } +C\quad$