Which of the following is incorrect? `int_(a c)^(b+c)f(x)dx=int_a^bf(x+c)dx` `int_(a c)^(b c)f(x)dx=int_a^bf(c x)dx` `int_(-a)^af(x)dx=1/2int_(-a)^a(f(x)+f(-x)dx` None of these

Options

A. (a) `int_(a+c)^(b+c)f(x)dx=int_(a)^(b)f(x+c)dx`
B. (b) `int_(ac)^(bc)f(x)dx=cint_(a)^(b)f(cx)dx`
C. (c) `int_(-a)^(a)f(x)dx=1/2int_(-a)^(a)f(x)+f(-x)dx`
D. (d) none of these

Answer

D

Explanation

`I_(1)=int_(a+b)^(b+c)f(x)dx`. Putting `x=t+c` and `dx=dt`, we get
`I_(1)=int_(a)^(b)f(t+c)dt=int_(a)^(b)f(x+c)dx`
Now `I_(2)=int_(ac)^(bc)f(x)dx`
Putting `x=tc` and `dx=cdt`, we get
`I_(2)=cint_(a)^(b)f(ct)dt=cint_(a)^(b)f(cx)dx`
`f(x)=1/2 (f(x)+f(-x)+f(x)-f(-x))`
`:. I_(3)=int_(-a)^(a)f(x)dx`
`=1/2 int_(-a)^(a)(f(x))+f(-x)+f(x)-f(-x)dx`
`=1/2int_(-a)^(a)(f(x)+f(-x))dx+1/2 int_(-a)^(a)(f(x)-f(-x))dx`
`=1/2 int_(-a)^(a)(f(x)+f(-x))dx`
as `f(x)+f(-x)` is even and `f(x)-f(-x)` is odd.