If `x^2!=n pi-1, n in N`. Then, the value of `int x sqrt((2sin(x^2+1)-sin2(x^2+1))/(2sin(x^2+1)+sin2(x^2+1)))dx` is equal to:

Options

A. `ln|(1)/(2)sec(x^(2)+1)|+C`
B. `ln|sec((x^(2)+1)/(2))|+C`
C. `(1)/(2)ln|sec(x^(2)+1)|+C`
D. `(1)/(2)ln|(2)/(sec(x^(2)+1))|+C`

Answer

B

Explanation

`I=(1)/(2)int2xsqrt((2sin(x^(2)+1)-sin2(x^(2)+1))/(2sin(x^(2)+1)+sin2(x^(2)+1)))dx`
`x^(2)+1=t rArr 2x = dt`
`I=(1)/(2)intsqrt((2sint-sin 2t)/(2sin t +sin 2r))dt`
`=(1)/(2)intsqrt((2-2cost)/(2+2cost))dt`
`=(1)/(2)int tan.(t)/(2)dt`
`=(1)/(2)(ln|sec.(t)/(2)|)/((1)/(2))+c`
`=ln|sec((x^(2)+1)/(2))|+c`