`f & g` are two real valued continuous functions and let `int g(x)dx = f^-1 (x) and f(x) = x^3 + x+ sin pix + 2` then the value of `int_2^4 xy (x) dx` is

Options

A. `(-11)/4-2/(pi)`
B. `11/4-2/(pi)`
C. `(-11)/4+2/(pi)`
D. `11/4+2/(pi)`

Answer

A

Explanation

Let `I=int_(2)^(4)xxg(x)dx`
Using by parts
`I=x.f^(-1)(x) int_(2)^(4)-int_(2)^(4)f^(-1)(x)dx`
`I=4f^(-1)(4)-2f^(-1)(2)-int_(2)^(4)f^(-1)(x)dx`
Now,
`f^(-1)(f(x))=x`
`f^(-1)(f(0))=0impliesf^(-1)(2)=0`
also `f^(-1)(f(1))=1gtf^(-1)(4)=1`
`int_(a)^(b)f(x) dx+int_(f(a))^(f(b))f^(-1)(x)dx=bf(b)-af(a)`
`int_(0)^(1)f(x) dx+int_(2)^(4)f^(-1)(x)dx=1.f(1)-0.f(0)`
`int_(0)^(1)(x^(3)+x+sinpix+2)dx+int_(2)^(4)f^(-1)(x)dx=4`
also `int_(2)^(4)f^(-1) (x)dx=4-((x^(4))/4+(x^(2))/2-(cospix)/(pi)+2x)_(0)^(1)`