Home
Class 12
MATHS
If f(a+b-x)=f(x), then prove that ...

If `f(a+b-x)=f(x),` then prove that `int_a^b xf(x)dx(a+b)/2int_a^bf(x)dxdot`

Text Solution

Verified by Experts

The correct Answer is:
NA
`I=int_(a)^(b)xf(x)dx=int_(a)^(b)xf(a+b-x)dx`
( `:'` given `f(x)=f(a+b-x))`
`=int_(a)^(b)(a+b-x)f((a+b)-(a+b-x))dx`
`=int_(a)^(b)(a+b-x)f(x)dx=(a+b)int_(a)^(b)f(x)dx-I`
or `2I=(a+b)int_(a)^(b)int_(a)^(b)f(x) dx` or `I=(a+b)/2 int_(a)^(b)f(x)dx`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.6|7 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.7|6 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.4|10 Videos

  • CURVE TRACING

    CENGAGE|Exercise Exercise|24 Videos

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

If f(a+b-x0=f(x) , then prove that int_a^b xf(x)dx=((a+b)/2)int_a^bf(x)dxdot

If f(a+b-x)=f(x), then prove that int_(a)^(b)xf(x)dx=(a+b)/(2)int_(a)^(b)f(x)dx

Prove that int_(a)^(b) f(x) dx= int_(a)^(b) f(a+b-x) dx

Prove that int_(a)^(b)f(x)dx=(b-a)int_(0)^(1)f((b-a)x+a)dx

If f(a+b-x)=f(x),\ t h e n\ int_a^b xf(x)dx is equal to (a+b)/2int_a^bf(b-x)dx b. (a+b)/2int_a^bf(b+x)dx c. (b-1)/2int_a^bf(x)dx d. (a+b)/2int_a^bf(x)dx

If f(a+b-x)=f(x) , then int_a^b x f(x)dx is equal to (A) (a-b)/2int_a^b f(a+b-x)dx (B) (a+b)/2int_a^b f(b-x)dx (C) (a+b)/2int_a^b f(x)dx (D) (b-a)/2int_a^b f(x)dx

int_a^b[d/dx(f(x))]dx

int _(a) ^(b) f (x) dx =

If f(a+b - x) = f(x) , then int_(a)^(b) x f(x) dx is equal to