Mathematics

# Evaluate the following : $\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \dfrac { x+\dfrac { \pi }{ 4 } }{ 2-\cos 2x } dx }$

##### SOLUTION
It is take $\displaystyle I = \int_{-\pi/4}^{\pi/4}\dfrac{x + \pi/4}{2-\cos 2x}dx$      ...(i)

use property $\displaystyle \int_a^b f(x) dx = \int_a^b f(a+b - x)dx$

Then

$I =\displaystyle \int_{-\pi/4}^{\pi/4} \dfrac{-x+\pi/4}{2-\cos 2x}dx$      ...(2)

adding (1) and (2) we have
$2I =\displaystyle \int_{-\pi/4}^{\pi/4}\dfrac{\pi/2}{2-\cos 2x}dx = \dfrac{\pi}{2} \int_{-\pi/4} ^{\pi/4} \dfrac{dx}{1 + 2\sin^2x}$

$I = \displaystyle \dfrac{\pi}{4}\int_{-\pi/4} ^{\pi/4} \dfrac{dx}{1+2\sin^2x} = \dfrac{\pi}{2} \int_0^{\pi/4} \dfrac{dx}{1 + 2\sin^2x}$

$I = \dfrac{\pi}{2} \displaystyle \int_{0}^{\pi/4} \dfrac{\sec^2xdx}{\sec^2x + 2\tan^2x} = \dfrac{\pi}{2} \int_0^{\pi/4} \dfrac{\sec^2x dx}{3\tan^2x + 1}$

Let $\sqrt{3} \tan x = t \Rightarrow \sec^2x\, dx = \dfrac{dt}{\sqrt{3}}$

$\left.\begin{matrix} I = \dfrac{\pi}{2\sqrt{3}} \displaystyle \int_0^{\sqrt{3}} \dfrac{dt}{t^2 + 1} = \dfrac{\pi}{2\sqrt{3}} \tan^{-1} t \end{matrix}\right|^{\sqrt{3}}_0$

$= \dfrac{\pi}{2\sqrt{3}} \times \dfrac{\pi}{3} = \dfrac{\pi^2}{6\sqrt{3}}$

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

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int \dfrac{dx}{e^x + e^{-x} +2}$ is equal to
• A. $\dfrac{1}{e^x+1} +C$
• B. $\dfrac{1}{1+e^{-x}} +C$
• C. $\dfrac{1}{e^{-x}-1} +C$
• D. $\dfrac{-1}{e^x+1} +C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following integral
$\int { \cfrac { \sin { \left( x-\alpha \right) } }{ \sin { \left( x+\alpha \right) } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
The value of $\dfrac{{\sin x + \cos x}}{{\sqrt {1 + \sin 2x} }}dx$ is
• A. $\sin x+c$
• B. $\cos x+c$
• C. $\dfrac{1}{2}(\sin x+\cos x)$
• D. $x+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Let $I_{1}=\int_{-2}^{2} \dfrac{x^{6}+3 x^{5}+7 x^{4}}{x^{4}+2} d x$ and$I_{2}=\int_{-3}^{1} \dfrac{2(x+1)^{2}+11(x+1)+14}{(x+1)^{4}+2} d x,$ then the value of$I_{1}+I_{2}$ is
• A. 8
• B. $200 / 3$
• C. None of these
• D. $100 / 3$

If $\displaystyle \int \dfrac{e^x - 1}{e^x + 1}dx =f(x) + c$ then $f(x) =$