Mathematics

$$\displaystyle\int^{\pi}_{-\pi}\dfrac{2x(1+\sin x)}{1+\cos^2x}dx$$.


SOLUTION
$$\displaystyle I = \int_{-\pi}^{\pi} \cfrac{2x}{1+\cos^2x}dx + \int_{-\pi}^{\pi} \cfrac{2x\sin x}{1+\cos^2x} dx$$
$$I = \displaystyle \int_0^{\pi} 2\cfrac{(\pi-x)\sin x}{1 + \cos^2x} dx$$
$$2I = \displaystyle 2\pi\int_0^{\pi}\cfrac{\sin x}{1+\cos^2x}$$
$$I = \displaystyle \pi \int_{-1}^1 \cfrac{dt}{1+t^2}$$ 
$$I = \displaystyle \pi \Bigg[\tan^{-1} t\Bigg]_{-1}^1 = \cfrac{\pi^2}{2}$$
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Hard
Write a value of 
$$\displaystyle\int { \cfrac { \sec ^{ 2 }{ x }  }{ { \left( 5+\tan { x }  \right)  }^{ 4 } }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
The value of $$\displaystyle \int_{-\pi }^{\pi }\left ( 1-x^{2} \right )\sin x\cos ^{2}xdx$$ is
  • A. $$\displaystyle \pi -\frac{\pi ^{3}}{3}$$
  • B. $$\displaystyle 2\pi -\pi ^{3}$$
  • C. $$\displaystyle \frac{7}{2}-2\pi ^{3}$$
  • D.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
$$\int { \cfrac { \csc ^{ 2 }{ x } -2005 }{ \cos ^{ 2005 }{ x }  } dx } $$ is equal to

  • A. $$\cfrac { \tan { x } }{ { \left( \cos { x } \right) }^{ 2005 } } +C$$
  • B. $$\cfrac { -\tan { x } }{ { \left( \cos { x } \right) }^{ 2005 } } +C$$
  • C. None of these
  • D. $$\cfrac { \cot { x } }{ { \left( \cos { x } \right) }^{ 2005 } } +C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
The value of 
$$\displaystyle \int{(1+(2tanx)(\tan x+\sec x))}^{1/2}$$ dx  is equal to
  • A. $$In\|sec x(\csc x\tan x)|+C$$
  • B. $$In\|sec x(\sec x\tan x)|+C$$
  • C. $$In\|cos x(\sec x\tan x)|+C$$
  • D. $$In\|sec x(\sec x+\tan x)|+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
Evaluate:
$$ \int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer