Mathematics

The value of the integral $$\displaystyle \int_{\frac{\pi}6}^{\frac{\pi}2} \left (\dfrac {1 + \sin 2x + \cos 2x}{\sin x + \cos x}\right )dx$$ is equal to


ANSWER

$$1$$


SOLUTION
Let 
$$I = \displaystyle \int_{\pi / 6}^{\pi /2} \left (\dfrac {1 + \sin 2x + \cos 2x}{\sin x + \cos x}\right )dx$$

$$=  \displaystyle \int_{\pi / 6}^{\pi /2} \left (\dfrac {1 + 2\sin x \cos x + 2\cos^{2} x - 1}{(\sin x + \cos x)}\right )dx$$

$$=  \displaystyle \int_{\pi / 6}^{\pi /2} \dfrac {2\cos x (\sin x + \cos x)}{(\sin x + \cos x)} dx$$

$$= \displaystyle  \int_{\pi / 6}^{\pi /2} 2\cos x dx = 2[\sin x]_{\pi / 6}^{\pi /2}$$

$$= 2\left (\sin \dfrac {\pi}{2} -\sin \dfrac {\pi}{6}\right ) = 2\left (1 - \dfrac {1}{2}\right ) = 2\times \dfrac {1}{2} = 1$$
View Full Answer

Its FREE, you're just one step away


Single Correct Hard Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Evaluate the following integrals:
$$\int { \cfrac { \cos { 2x }  }{ \sqrt { \sin ^{ 2 }{ 2x } +8 }  }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
$$\displaystyle \int\sqrt{\frac{x-1}{2x-3}}dx=$$
  • A. $$sin^{-1}x+\sqrt{1-x^{2}}+c$$
  • B. $$sin^{-1}x-\sqrt{1-x^{2}}+c$$
  • C. $$2sin^{-1}x-\sqrt{1-x^{2}}+c$$
  • D. $$\dfrac{1}{2}\displaystyle \sqrt{2x^{2}-5x+3}+\frac{1}{4}\cosh^{-1}(4x-5)+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium

$$\displaystyle \int_{0}^{a}[f(a+x)+f(a-x)]dx_{=}$$
  • A. $$\displaystyle \int_{0}^{a}f(2x)dx$$
  • B. $$\displaystyle \int_{0}^{a}f(x)dx$$
  • C. $$\displaystyle \int_{-\mathrm{a}}^{\mathrm{a}}\mathrm{f}(\mathrm{x})$$ dx
  • D. $$\displaystyle \int_{0}^{2a}f(x)dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Hard
Evaluate:
$$\displaystyle\int _{ 0 }^{ \pi /2 }{ \log { \left( \sin { x }  \right) dx }  } $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
solve $$\int \frac{1}{1+e^{-1}}\;dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer