Mathematics

Find m if :$$\displaystyle \int_{0}^{\pi}\left ( \dfrac{\sqrt{1+\cos\ 2x}}{2} \right )dx=\sqrt m$$


ANSWER


SOLUTION
$$\displaystyle\int_{0}^{\pi}{\left(\dfrac{\sqrt{1+\cos{2x}}}{2}\right)dx}$$
$$=\displaystyle\int_{0}^{\pi}{\left(\dfrac{\sqrt{1+2{\cos}^{2}{x}-1}}{2}\right)dx}$$
$$=\dfrac{\sqrt{2}}{2}\displaystyle\int_{0}^{\pi}{\cos{x}dx}$$
$$=\dfrac{\sqrt{2}}{2}\left[\sin{x}\right]_{0}^{\pi}$$
$$=\dfrac{\sqrt{2}}{2}\left[\sin{\pi}-\sin{0}\right]=0$$
$$\therefore\,m=0$$
View Full Answer

Its FREE, you're just one step away


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

Realted Questions

Q1 Single Correct Hard
$$\int { \cfrac { 3{ x }^{ 5 } }{ \left( 1+{ x }^{ 12 } \right)  }  } dx=$$?
  • A. $$\tan ^{ -1 }{ { x }^{ 6 } } +C$$
  • B. $$\cfrac { 1 }{ 4 } \tan ^{ -1 }{ { x }^{ 6 } } +C$$
  • C. none of these
  • D. $$\cfrac { 1 }{ 2 } \tan ^{ -1 }{ { x }^{ 6 } } +C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
$$\int { \dfrac { \sqrt { 4+{ x }^{ 2 } }  }{ { x }^{ 6 } } dx } =\dfrac { { \left( a+{ x }^{ 2 } \right)  }^{ 3/2 }\left( { x }^{ 2 }-b \right)  }{ 120{ x }^{ 5 } }+C$$ then $$a+b$$ equals to:

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
Evaluate $$\displaystyle\int{\sqrt{x+\sqrt{x^2+2}}dx}$$.
  • A. $$\displaystyle\frac{1}{3}{(x+\sqrt{x^2+2})}^{\tfrac{1}{2}}+2\frac{1}{\sqrt{x+\sqrt{x^2+2}}}+C$$
  • B. $$\displaystyle\frac{2}{3}{(x+\sqrt{x^2+2})}^{\tfrac{2}{3}}+2\frac{1}{\sqrt{x+\sqrt{x^2+2}}}+C$$
  • C. None of these
  • D. $$\displaystyle\frac{1}{3}{(x+\sqrt{x^2+2})}^{\tfrac{3}{2}}-2\frac{1}{\sqrt{x+\sqrt{x^2+2}}}+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
Evaluate:
$$\displaystyle \int _{ 0 }^{ \sqrt { 2 }  }{ { x }^{ 2 } } dx$$ 
  • A. $$2-\sqrt { 2 } $$
  • B. $$2+\sqrt { 2 } $$
  • C. $$\dfrac{4} {3}$$
  • D. $$\dfrac{2\sqrt{2}}{3}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
If $$y=\displaystyle\int \dfrac {dx}{(1+x^{2})^{\frac {1}{2}}}$$ and $$y=0$$ when $$x=0$$, then value of $$y$$ when$$x=1$$, is:
  • A. $$\ln(2)$$
  • B. $$\ln(\sqrt{2})$$
  • C. $$\dfrac {1}{\sqrt {2}}$$
  • D. $$\ln(1+\sqrt{2})$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer