Mathematics

Let $$f:R\rightarrow R$$ and $$g:R\rightarrow R$$  be continuous functions, then the value of $$\displaystyle\int_{\displaystyle-\frac{\pi}{2}}^{\displaystyle\frac{\pi}{2}}{(f(x)+f(-x))(g(x)-g(-x))dx}$$, is equal to


ANSWER

$$0$$


SOLUTION
Let $${ I }=\int _{ -\cfrac { \pi }{ 2 } }^{ \cfrac { \pi }{ 2 } }{ \left( f\left( x \right) +f\left( -x \right) \right) \left( g\left( x \right) -g\left( -x \right) \right) dx } $$ ...(1)
Using property $$\int _{ a }^{ b }{ f\left( x \right) dx } =\int _{ a }^{ b }{ f\left( a+b-x \right) dx } $$
$${ I }=\int _{ -\cfrac { \pi }{ 2 } }^{ \cfrac { \pi }{ 2 } }{ \left( f\left( -x \right) +f\left( x \right) \right) \left( g\left( -x \right) -g\left( x \right) \right) dx } $$ ...(2)
Adding (1) and (2), we get
$$2I=\int _{ -\cfrac { \pi }{ 2 } }^{ \cfrac { \pi }{ 2 } }{ \left( f\left( -x \right) +f\left( x \right) \right) \left( \left( g\left( x \right) -g\left( -x \right) \right) +\left( g\left( -x \right) -g\left( x \right) \right) \right) dx } \\ =0\Rightarrow I=0$$
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 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
$$I = \int \dfrac {1}{x(x^{6}+1)}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Solve $$\displaystyle \int \cos^{3}x\ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Prove that : $$\displaystyle \int \dfrac {1}{a^{2} - x^{2}}dx = \dfrac {1}{2a}\ln \left |\dfrac {a + x}{a - x}\right | + c$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate the definite integral   $$\displaystyle \int_4^5e^xdx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Hard
$$\displaystyle \int_{\frac{5}{2}}^{5}\frac{\sqrt{(25-x^2)^3}}{x^4}\:dx$$ is equal to
  • A. $$\displaystyle \frac{\pi }{6}$$
  • B. $$\displaystyle \frac{2\pi }{3}$$
  • C. $$\displaystyle \frac{5\pi }{6}$$
  • D. $$\displaystyle \frac{\pi }{3}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer