Home
Class 12
MATHS
If f and g are two continuous functions ...

If f and g are two continuous functions being even and and odd, respectively, then `int_(-a)^(a)(f(x))/(b^(g(x)+1))dx` is equal to (a being any non-zero number and b is positive real number, `b ne 1` )

A

independent of f

B

independent of g

C

independent of both f and g

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int_(-a)^(a)(f(x))/(b^(g(x))+1)dx" (i)"`
`=int_(a)^(a)(f(-x))/(b^(-g(x))+1)dx`
`=int_(-a)^(a)(f(x))/(b^(-g(x))+1)dx`
`=int_(-a)^(a)(f(x).b^(g(x)))/(1+b^(g(x)))dx`
`therefore" "I=int(-a)^(a)(f(x).b^(g(x)))/(1+b^(g(x)))dx" (ii)"`
Adding (i) and (ii), `2I=int_(-a)^(a)(f(x))/(1+b^(g(x)))dx+int_(-a)^(a)(f(x).b^(g(x)))/(1+b^(g(x)))dx`
`=int_(-a)^(a)f(x)dx`
`therefore" "I=int_(0)^(a)f(x)dx`,
which is independent of g(x).
Promotional Banner

Similar Questions

Explore conceptually related problems

If f(x) and g(x) are two continuous functions defined on [-a,a], then the value of int_(-a)^(a){f(x)+f(-x)}{g(x)-g(-x)}dx is

Let f be an odd function then int_(-1)^(1) (|x| +f(x) cos x) dx is equal to

If f(x) and g(x) are continuous functions satisfying f(x) = f(a-x) and g(x) + g(a-x) = 2 , then what is int_(0)^(a) f(x) g(x)dx equal to ?

Let f and g be continuous functions on [ 0 , a] such that f(x)=f(x)=f(a-x)andg(x)+g(a-x)=4 , then int_(0)^(a) f(x)g(x) dx is equal to

Let f and g be continuous fuctions on [0, a] such that f(x)=f(a-x)" and "g(x)+g(a-x)=4 " then " int_(0)^(a)f(x)g(x)dx is equal to

If f and g are continuous functions on [0,pi] satisfying f(x)+f(pi-x)=g(x)+g(pi-x)=1 then int_(0)^( pi)[f(x)+g(x)]backslash dx is equal to