Mathematics

# Evaluate$\displaystyle \int _{ 0 }^{ \pi /2 }{ \dfrac { x+\sin { x } }{ 1+\cos { x } } dx }$

##### SOLUTION
$\begin{array}{l} \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { x+\sin x } }{ { 1+\cos x } } dx } \\ =\, \, \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { x+2\sin \frac { x }{ 2 } \cos \frac { x }{ 2 } } }{ { 2{ { \cos }^{ 2 } }\frac { x }{ 2 } } } dx } \\ =\, \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( { x\frac { 1 }{ 2 } { { \sec }^{ 2 } }\frac { x }{ 2 } +\tan \frac { x }{ 2 } } \right) }\, \, dx \\ \therefore \, \, \, from\, formula\, ,\int { \left[ { x\, { f^{ ' } }(x)+f(x) } \right] } \, \, dx=x\, f(x) \\ =\, \left[ { x\tan \frac { x }{ 2 } } \right] _{ 0 }^{ \frac { \pi }{ 2 } } \\ =\, \left( { \frac { \pi }{ 2 } \tan \frac { \pi }{ 4 } } \right) -0 \\ =\frac { \pi }{ 2 } \end{array}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 Single Correct Medium
$\int_0^1 {\dfrac{{dx}}{{x + \sqrt x }}}$ equals
• A. $ln\ 2$
• B. $2\ln 2-2$
• C. $2ln\ 2-1$
• D. $2ln\ 2$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Find :
$\int { \dfrac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int { \cfrac { \cos { x } -1 }{ \sin { x } +1 } } { e }^{ x }dx$ is equal to:
• A. $c-\cfrac { { e }^{ x }\sin { x } }{ 1+\sin { x } }$
• B. $c-\cfrac { { e }^{ x }}{ 1+\sin { x } }$
• C. $c-\cfrac { { e }^{ x }\cos { x } }{ 1+\sin { x } }$
• D. $\cfrac { { e }^{ x }\cos { x } }{ 1+\sin { x } } +c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
37 If $n > 1,$ and $\displaystyle I=\int _{0}^{\infty} \frac{dx}{(x+\sqrt{1+x^{2}})^{n}}$ then $I$  equals
• A. $\displaystyle \frac{2n}{n^{2}-1}$
• B. $\displaystyle \frac{n}{2(n^{2}-1)}$
• C. $\displaystyle \sqrt{ n^{2}-1}$
• D. $\displaystyle \frac{n}{n^{2}-1}$

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 $\displaystyle f(x)>0 \forall x \in R$