Mathematics

The value of $$\displaystyle \int_{-\pi /2}^{\pi /2}\frac{dx}{\sin \:^{3}x+\sin x}$$is


ANSWER

$$0$$


SOLUTION
Let $$\displaystyle I=\int _{ -\dfrac { \pi  }{ 2 }  }^{ \dfrac { \pi  }{ 2 }  }{ \dfrac { dx }{ \sin ^{ 3 }{ x } +\sin { x }  }  } $$
Using $$\int _{ a }^{ b }{ f\left( x \right)  } dx=\int _{ a }^{ b }{ f\left( a+b-x \right)  } dx$$
$$\displaystyle I=\int _{ -\dfrac { \pi  }{ 2 }  }^{ \dfrac { \pi  }{ 2 }  }{ \dfrac { dx }{ \sin ^{ 3 }{ \left( -x \right)  } +\sin { \left( -x \right)  }  }  } dx$$
$$\displaystyle =-\int _{ -\dfrac { \pi  }{ 2 }  }^{ \dfrac { \pi  }{ 2 }  }{ \dfrac { dx }{ \sin ^{ 3 }{ x } +\sin { x }  }  } dx=-I\\ \therefore 2I=I\Rightarrow I=0$$
View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

Realted Questions

Q1 Subjective Hard
Evaluate the following integrals:-
$$\int {\dfrac{{x + 1}}{{\sqrt {{x^2} - x + 1} }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Evaluate the following integral:
$$\displaystyle \int { \cfrac { 3{ x }^{ 5 } }{ 1+{ x }^{ 12 } }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate $$\displaystyle\int^1_0\dfrac{dx}{\sqrt{1-x^2}}$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Hard
Evaluate the given integral.
$$\int {x.{\sin}^{-1}{x}} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Consider two differentiable functions $$f(x), g(x)$$ satisfying $$\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$$ & $$\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$$. where $$\displaystyle f(x)>0    \forall  x \in  R$$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives


1 Verified Answer | Published on 17th 08, 2020

View Answer