AP EAMCET20225 Jul 2022Evening ShiftMathematicsDefinite IntegrationActual
_ - ^ 2 x(1+ x) 1+ ^2 x d x=
Options
- A2
- B^2
- C+2
- D/ 2
Correct answer
B. ^2
Step-by-step solution
Let I= _ - ^ [ 2 x(1+ x) (1+ ^2 x ) ] d x= _ - ^ [ 2 x (1+ ^2 x ) ] d x+ _ - ^ [ 2 x( x) (1+ ^2 x ) ] d xI=I₁+I₂ Take I₁= _ - ^ [ 2 x (1+ ^2 x ) ] d x aligned & f(x)= [ 2 x 1+ ^2 x ] & f(-x)= [ 2(-x) 1+ ^2( - x ) ] aligned aligned & =- [ 2 ( ^x ) 1+(- x)^2 ] & =- [ 2 ( ^x ) 1+ ^2 x ] aligned =-f ( ^x ) I₁= _ - ^ 2 x 1+ ^2 x d x=0 ( since f(x) is an odd function) Again consider aligned & g ( X )= 2 x x 1+ ^2 x & g (- x )= 2(- x ) (- x ) 1+ ^2(- x ) aligned = 2( x ) ( x ) 1+ ^2( x ) [ since (-x)=- x, (-x)= x]= g ( ^