Mathematics

# $\displaystyle \int_{\pi /4}^{3\pi /4}\frac{dx}{1+\cos x}$ is equal to

$2$

##### SOLUTION
Let $\displaystyle I=\int { \dfrac { 1 }{ 1+\cos { x } } } dx$

Substitute $\displaystyle t=\tan { \dfrac { x }{ 2 } } \Rightarrow dt=\dfrac { 1 }{ 2 } dxsec^{ 2 }x$

$\displaystyle I=\int { \dfrac { 2 }{ \left( { u }^{ 2 }+1 \right) \left( \dfrac { 1-{ u }^{ 2 } }{ { u }^{ 2 }+1 } +1 \right) } } du=\int { du } =u=\tan { \dfrac { x }{ 2 } }$

$\displaystyle \therefore \int _{ \dfrac { \pi }{ 4 } }^{ \dfrac { 3\pi }{ 4 } }{ \dfrac { dx }{ 1+\cos { x } } } =\left[ \tan { \dfrac { x }{ 2 } } \right] _{ \dfrac { \pi }{ 4 } }^{ \dfrac { 3\pi }{ 4 } }=2$

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

#### Realted Questions

Q1 Subjective Medium
Evaluate the following integral:
$\displaystyle \int { \cfrac { \left( x+1 \right) { e }^{ x } }{ \sin ^{ 2 }{ \left( x{ e }^{ x } \right) } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The value of $\displaystyle \int_{-\pi /2}^{\pi /2}\sqrt{\frac{1}{2}\left ( 1-\cos 2x \right )}$ dx is
• A. $0$
• B. $\displaystyle \frac{1}{2}$
• C. None of these.
• D. $2$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\displaystyle\int^1_0\dfrac{dx}{(1+x^2)}$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following : $\displaystyle\int \dfrac{1}{\sqrt{11-4x^{2}}}.dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
In calculating a number of integrals we had to use the method of integration by parts several times in succession.
The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts
$\displaystyle \int u\left ( x \right )v\left ( x \right )dx=u\left ( x \right )v_{1}-u'\left ( x \right )v_{2}\left ( x \right )+u''\left ( x \right )v_{3}\left ( x \right )+...+\left ( -1 \right )^{n-1}u^{n-1}\left ( x \right )V_{n}\left ( x \right ) \\ -\left ( -1 \right )^{n-1}\int u^{n}\left ( x \right )V_{n}\left ( x \right )dx$
where  $\displaystyle v_{1}\left ( x \right )=\int v\left ( x \right )dx,v_{2}\left ( x \right )=\int v_{1}\left ( x \right )dx ..., v_{n}\left ( x \right )= \int v_{n-1}\left ( x \right )dx$
Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when  calculating $\displaystyle \int P_{n}\left ( x \right )Q\left ( x \right )dx,$ where $\displaystyle P_{n}\left ( x \right )$ is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n+1 times.