Mathematics

# Solve: $I=\int _{ o }^{ \pi }{ \dfrac { xdx }{ 1+\sin { x } } }$

##### SOLUTION
$I=\displaystyle \int_0^n \dfrac {x\ dx}{1+\sin^2x}$
$=\displaystyle \int _0^n \dfrac {x}{\cos^2x / (1-\sin x)}$
$=\displaystyle \int \dfrac {x}{\cos^2 x}-\dfrac {x \sin x}{\cos ^2 s}dx$
$I_1=2.=\displaystyle \int \dfrac {x}{1+\cos 2x}$
$2=\displaystyle \int \dfrac {1}{2} x\tan x-\displaystyle \int \dfrac {1}{2} \tan \ dx$
$x\tan x- (-\ln |\cos x|)$
$I_2 =\displaystyle \int \dfrac {x \sin x}{\cos ^2 x} dx$
$=x \sin x \tan x-\displaystyle \int (\sin x+x \cos x)\tan x\ dx$
$\displaystyle \int \tan x \sin x dx+\displaystyle \int x \tan x \cos dx$
$\Rightarrow \ (\ln |\tan x +\sec x|-\sin x) -(x \cos x +\sin x)$
$x \tan x+\ln |\cos x|-(x\sin x \tan x+x \cos x-\ln |\tan x+\sec x|)$
$\Rightarrow \ \tan x+\ln |\cos x|-x\sin x \tan x -x\cos x +\ln (\tan x+\sec x)+C$
$\Rightarrow \ =\displaystyle \lim _{x+0} \to 0$
$=\displaystyle \lim _{x+n}\Rightarrow \ \pi$
$\therefore \ \pi -0\ , \ I=\pi$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
Evaluate:
$\displaystyle\int { \frac { { x }^{ 4 }+4 }{ { x }^{ 2 }-2x+2 } } dx$
• A. $\dfrac{x^3}{2}+x^2+2x+C$
• B. $\dfrac{x^3}{3}+x^2+x+C$
• C. $\dfrac{x^3}{3}+x^2-2x+C$
• D. $\dfrac{x^3}{3}+x^2+2x+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int\frac{d_{X}}{\sin^{2}x+\sin2x}=$
• A. $\displaystyle \frac{-1}{2}\log|\frac{\tan x}{\tan x+2}|+c$
• B. $\displaystyle \frac{1}{3}\log|\displaystyle \frac{\tan x}{\tan x+2}|+c$
• C. $\displaystyle -\frac{1}{3}\log|\displaystyle \frac{\tan x}{\tan x+2}|+c$
• D. $\displaystyle \frac{1}{2}\log|\displaystyle \frac{\tan x}{\tan x+2}|+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int\{\frac{(\log x-1)}{1+(\log x)^{2}}\}^{2}dx$ is equal to
• A. $\displaystyle \frac{xe^{x}}{1+x^{2}}+c$
• B. $^{\dfrac{x}{x^{2}+1}+c}$
• C. $\displaystyle \frac{\log x}{(\log x)^{2}+1}+c$
• D. $\displaystyle \frac{x}{(\log x)^{2}+1}+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\displaystyle\int x\left(\dfrac{\sec 2x-1}{\sec 2x+1}\right)dx$.

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
Integrate:
$\int _{ 0 }^{ \infty }{ \dfrac { x\tan ^{ -1 }{ x } }{ { (1+{ x }^{ 2 }) }^{ 2 } } } dx$ equals ?
• A. $\pi/2$
• B. $\pi/6$
• C. $\pi/4$
• D. $\pi/8$