Mathematics

Evaluate the following integral
$$\int { \sqrt { \cfrac { 1-\cos { x }  }{ 1+\cos { x }  }  }  } $$


SOLUTION
$$I=\displaystyle\int{\sqrt{\dfrac{1-\cos{x}}{1+\cos{x}}}dx}$$

$$I=\displaystyle\int{\sqrt{\dfrac{1-1+2{\sin}^{2}{\dfrac{x}{2}}}{1+2{\cos}^{2}{\dfrac{x}{2}}-1}}dx}$$

$$I=\displaystyle\int{\sqrt{\dfrac{{\sin}^{2}{\dfrac{x}{2}}}{{\cos}^{2}{\dfrac{x}{2}}}}dx}$$

$$I=\displaystyle\int{\dfrac{\sin{\dfrac{x}{2}}}{\cos{\dfrac{x}{2}}}dx}$$

Let $$t=\cos{\dfrac{x}{2}}\Rightarrow\,dt=\dfrac{-1}{2}\sin{\dfrac{x}{2}}dx$$

$$\Rightarrow\,-2dt=\sin{\dfrac{x}{2}}dx$$

$$I=-2\displaystyle\int{\dfrac{dt}{t}}$$

$$\Rightarrow\,I=-2\log{\left|t\right|}+c$$

$$\therefore\,I=-2\log{\left|\cos{\dfrac{x}{2}}\right|}+c$$ where $$t=\cos{\dfrac{x}{2}}$$

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 84
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
The value of $$\displaystyle\int _{ 0 }^{ \infty  }{ \dfrac { dx }{ \left( { x }^{ 2 }+4 \right) \left( { x }^{ 2 }+9 \right)  }  } $$ is
  • A. $$\dfrac { \pi }{ 20 } $$
  • B. $$\dfrac { \pi }{ 40 } $$
  • C. $$\dfrac { \pi }{ 80 } $$
  • D. $$\dfrac { \pi }{ 60 } $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Hard
$$\displaystyle \int  \cfrac{5 x-2}{1+2 x+3 x^{2}} dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
Evaluate:
$$\displaystyle \int_{\pi/4}^{3\pi/4} \dfrac{dx}{1+\cos x}$$ 
  • A. $$-2$$
  • B. $$\dfrac 1 2$$
  • C. $$\dfrac {-1} 2$$
  • D. $$2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
The value of $$\displaystyle \int _{  {- \pi  }/{ 4 }  }^{ { \pi  }/{ 4 }  }{ \sin ^{ 103 }{ x } .\cos ^{ 101 }{ x }  } dx$$ is
  • A. $${ \left( \cfrac { \pi }{ 4 } \right) }^{ 103 }\quad $$
  • B. $${ \left( \cfrac { \pi }{ 4 } \right) }^{ 101 }\quad $$
  • C. $$2$$
  • D. $$0$$

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