Mathematics

# $\int \dfrac {x+\sin }{1+\cos x}dx=$

##### ANSWER

$x\ \tan \dfrac {x}{2}+c$

##### SOLUTION
$\displaystyle \int \dfrac {x+\sin x}{(1+\cos x)}dx=\displaystyle \int \dfrac {x+2 \sin (x/2)\cos (x/2)}{1+2\cos^2 (x/2)-1}dx$
$\displaystyle \int \dfrac {x+2\sin \left (\dfrac {x}{2}\right) \cos \dfrac {x}{2}}{2\cos^2x/2}dx =\displaystyle \int \dfrac {xdx}{2\cos^2 (x/2)}+\displaystyle \int tan \dfrac {x}{2}dx$
$\dfrac { 1 }{ 2 } \int { x } \sec ^{ 2 }{ \left( \dfrac { x }{ 2 } \right) dx } +\int { \tan { \left( \dfrac { x }{ 2 } \right) } dx }$
integrate first integral by parts
$U=x$
$du=dx$
$du=\dfrac {1}{2}\sec^2 x/2 dx$
$u=\tan x/2$
$\dfrac {1}{2} \displaystyle \int x\sec^2 \left (\dfrac {x}{2}\right)dx +\displaystyle \int \tan \dfrac {x}{2}dx =x \tan \dfrac {x}{2}-\displaystyle \int \tan \dfrac {x}{2}dx +\displaystyle \int \tan \dfrac {x}{2}dx$
$=x\tan (x/2)+c$

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

#### Realted Questions

Q1 Assertion & Reason Hard
##### ASSERTION

Assume: $\displaystyle I=\int \frac{\sqrt{\cos 2x}}{\sin x}dx$

$\displaystyle I=\ln \left [ \left ( \frac{1-\sqrt{1-\tan ^{2}x}}{\tan x} \right )\left ( \frac{\sqrt{2}+\sqrt{1-\tan ^{2}x}}{\sqrt{2}-\sqrt{1-\tan ^{2}x}} \right )^{\frac{1}{\sqrt{2}}} \right ]+C$

##### REASON

$\displaystyle \tan x=\sin \theta \rightarrow I=\int \frac{\cos ^{2}\theta d\theta }{\sin \theta \left ( 1+\sin ^{2}\theta \right )}$

• A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The value of $\displaystyle \int\limits_0^1 {\dfrac{{x{{\tan }^{ - 1}}x}}{{{{\left( {1 + {x^2}} \right)}^{3/2}}}}} dx$ is
• A. $\dfrac{{4 + \pi }}{{4\sqrt 2 }}$
• B. $\dfrac{\pi }{2}$
• C. $- \dfrac{\pi }{2}$
• D. $\dfrac{{4 - \pi }}{{4\sqrt 2 }}$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate : $\displaystyle\int _ { 1 } ^ { 2 } \frac { \ln x } { x ^ { 2 } } d x$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int \dfrac{1}{1+e^x}dx$ is equal to?
• A. $log_e\left(\dfrac{e^x+1}{e^x}\right)+c$
• B. $log_e\left(\dfrac{e^x-1}{e^x}\right)+c$
• C. $log_e\left(\dfrac{e^x}{e^x-1}\right)+c$
• D. $log_e\left(\dfrac{e^x}{e^x+1}\right)+c$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Medium
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 $\displaystyle f(x)>0 \forall x \in R$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives

1 Verified Answer | Published on 17th 08, 2020