Mathematics

# The value of integral $\displaystyle \int _{ \pi /4 }^{ 3\pi /4 }{ \frac { x }{ 1+\sin x } dx }$ is

$\pi (\sqrt { 2 } -1)$

##### SOLUTION
Let us take $I = \displaystyle \int_{\pi/4}^{3\pi/4} \dfrac{x}{1 + \sin x} dx$ ___(1)

we can write

$I = \displaystyle \int_{\pi/4}^{3 \pi/4} \dfrac{\pi - x}{1 + \sin x} dx$ ___(2)

adding (1) and (2) we get

$2I = \displaystyle \int_{\pi/4}^{3 \pi/4} \dfrac{\pi}{1 + \sin x} dx \Rightarrow I = \dfrac{\pi}{2} \int_{\pi/4}^{3\pi/4} \dfrac{dx}{1 + \sin x}$

$I = \dfrac{\pi}{2} \displaystyle \int_{\pi/4}^{3 \pi/4} \dfrac{1 - \sin x}{\cos^2 x} dx$

$I = \dfrac{\pi}{2} \displaystyle \int_{\pi/4}^{3\pi/4} \sec^2 x - \sec x \tan x \, dx$

$I = \dfrac{\pi}{2} \tan x - \sec x |_{\pi/4}^{3\pi/4}$

$= \dfrac{\pi}{2} (-1 + \sqrt{2} - 1 + \sqrt{2}) = \pi (\sqrt{2} - 1)$
option (B)

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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
$\int\limits_1^2 {{e^x}\left( {\frac{1}{x} - \frac{1}{{{x^2}}}} \right)} dx\,\,is\,\,equal\,\,to:$
• A. $e\left( {e - 1} \right)$
• B.
• C. none of these
• D. $e\left( {\frac{e}{2} - 1} \right)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Solve:
$\displaystyle \int_{1}^{\sqrt{3}} \cfrac{d x}{1+x^{2}} \text { equals }$
• A. $\dfrac {\pi}{3}$
• B. $\dfrac {2\pi}{3}$
• C. $\dfrac {\pi}{6}$
• D. $\dfrac {\pi}{12}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int {\sqrt {x + a\sqrt {ax - {a^2}} } \,dx,0 < a < 2 = \frac{2}{{{a^{\frac{3}{2}}}}}{{\left\{ {ax + a\sqrt {ax - {a^2}} } \right\}}^{\frac{3}{2}}} - \frac{{\sqrt a }}{2}\left[ {A + B} \right]} + c$. Then
• A. $A = \left[ {\left\{ {\left( {\sqrt {ax - {a^2}} + \frac{{{a^2}}}{2}} \right)\sqrt {ax + {a^2}\sqrt {ax - {a^2}} } } \right\}} \right]$
• B. $A = \left\{ {\left( {\sqrt {ax - {a^2}} + \frac{{{a^2}}}{2}} \right)\sqrt {ax + a\sqrt {ax - {a^2}} } } \right\}$
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1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate:
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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 $\displaystyle f(x)>0 \forall x \in R$