Mathematics

# If $\int \dfrac {1}{1 + \sin x}dx = \tan \left (\dfrac {x}{2} + a\right ) + b$, then

$a = \dfrac {\pi}{4}, b\in R$

##### SOLUTION
$\int \dfrac{1}{1+\sin x}dx= \tan \left(\dfrac{x}{2}+a\right)+b$
differentiating both sides wrt to x, we get :
$\dfrac{1}{1+\sin x}=\dfrac{d}{dx}\left(\tan \left(\dfrac{x}{2}+a\right)+b\right)$

$\therefore \dfrac{1}{1+\sin x}= \sec^{2}\left(\dfrac{x}{2}+a\right)\times \dfrac{1}{2}$

$\therefore \dfrac{2}{1+\sin x}=\sec^{2}\left(\dfrac{x}{2}+a\right)$

$\therefore 1+\sin x=2 \cos^{2}\left(\dfrac{x}{2}+a\right)$

$\therefore \sin x= \cos (x+2a)$

$\therefore \cos \left(\dfrac{\pi }{2}-x\right)=\cos (x+2a)$

$\therefore \dfrac{\pi }{2}-2a=2x$

$\therefore \dfrac{\pi }{2}-2a=0$

$\therefore a^{2}=\dfrac{\pi }{2\times 2}=\dfrac{\pi }{4}$

$\therefore a=\dfrac{\pi }{4},b=R$

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

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int \sqrt {2x+5}dx$
• A. $\dfrac 1{\sqrt {2x+5}}$
• B. $\dfrac{3}{5}(2x+5)^{3/5}$
• C. $\dfrac 1{2x+5}$
• D. $\dfrac 13(2x+5)^{3/2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Integral of $\displaystyle \frac{(4x^{2}-2\sqrt{x})}{x} +\frac{1}{1+x^{2}}-5 co\sec ^{2}x$ is

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the given integral: $\displaystyle\int_{0}^{1}(1-x^{2})dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle \int { \cfrac { { x }^{ 3 } }{ \sqrt { 1+x^2 } } } dx$
• A. $\sqrt { 1+x } -\cfrac { x }{ 3 } { \left( 1+{ x }^{ 2 } \right) }^{ 3/2 }+c$
• B. $x\sqrt { 1+{ x }^{ 2 } } +\cfrac { 2 }{ 3 } { \left( 1+{ x }^{ 2 } \right) }^{ 3/2 }+c$
• C. ${ x }^{ 2 }\sqrt { 1+{ x }^{ 2 } } -\cfrac { 1 }{ 3 } { \left( 1+{ x }^{ 2 } \right) }^{ 3/2 }+c$
• D. $\dfrac{{ x }^{ 2 }\sqrt { 1+{ x }^{ 2 } }}{3}-\cfrac { 2 }{ 3 } {\sqrt{1+{ x }^{ 2 }} }+c$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$