Mathematics

# The integral $\displaystyle \int _{ \dfrac { \pi }{ 4 } }^{ \dfrac { 3\pi }{ 4 } }{ \dfrac { dx }{ 1+\cos { x } } }$

$2$

##### SOLUTION
Multiply Numerator & denominator by $(1 - cos x)$

$\displaystyle \int \dfrac{1 - \cos x}{(1 + \cos x) (1 - \cos x)}dx = \int \dfrac{1 -\cos x}{1 -\cos x}dx$

$\displaystyle = \int \dfrac{1 - \cos x}{\sin ^2 x}dx = \int \dfrac{1}{\sin^2 x}dx - \int \dfrac{\cos x}{\sin^2 x}dx$

$\displaystyle = - \cot x - \int \dfrac{1}{y^2}dy \, (y = \sin x , dy = \cos x \, dx)$

$\displaystyle = -\cot x + \dfrac{1}{y} = \left[-\cot x + \dfrac{1}{\sin x} + c\right]_{\frac{\pi}{4}}^{^3\frac{\pi}{4}}$

$\displaystyle = \left[-\frac{1}{\tan x} + \frac{1}{\sin x} \right]^{^3\frac{\pi}{4}} - \left[-\frac{1}{\tan x} + \frac{1}{\sin x}\right]^{^3\frac{\pi}{4}}$

$\displaystyle = (1 + \sqrt 2) - (-1 + \sqrt 2) = 2$

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

#### Realted Questions

Q1 Single Correct Medium
What is $\int_ {-\frac{\pi}{2}}^{\frac{\pi}{2}} x \, sin \, x dx$equal to ?
• A.
• B. -2
• C. $\pi$
• D. 2

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\int { f\left( x \right)dx=f\left( x \right) } ,$ then $\int { \left\{ f\left( x \right) \right\}^2 }$ dx is equal to :
• A. ${ \left\{ f\left( x \right) \right\}^3 }$
• B. $\dfrac { { \left\{ f\left( x \right) \right\} ^{ 3 } } }{ 3 }$
• C. ${ \left\{ f\left( x \right) \right\}^2 }$
• D. $\frac { 1 }{ 2 } \left\{ f\left( x \right) \right\}^2$

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
If $I_1=\displaystyle\int^1_x\dfrac{dt}{1+t^2}$ and $I_2=\displaystyle\int^{1/x}_1\dfrac{dt}{1+t^2}$ for $x > 0$, then?
• A. $I_1 > I_2$
• B. $I_2 > I_1$
• C. $I_1 = I_2$
• D. $I_2=\cot^{-1}x-\pi/4$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int e^{x}(\frac{1-x}{1+x^{2}})^{2}$ dx $=$
• A. $\displaystyle \frac{-e^{x}}{(1+x)^{2}}+c$
• B. $\displaystyle \frac{e^{x}}{(1+x^{2})^{2}}+c$
• C. $\displaystyle \frac{-e^{x}}{(1+x^{2})^{2}}+c$
• D. $\displaystyle \frac{e^{x}}{1+x^{2}}+c$

Let $\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$