MHT CET20249 May 2024Morning ShiftMathematicsDefinite IntegrationActual
The value of I=I= _ - 2 ^ 2 x^2 x 1+e^ -x ~d x is equal to
Options
- A^2 4 -2
- B^2 4 +2
- C^2- e ^ 2
- D^2+ e ^ 2
Correct answer
A. ^2 4 -2
Step-by-step solution
array r I= _ - 2 ^ 2 x^2 x 1+e^ -x ~d x ...(i) I= _ - 2 ^ 2 ( - 2 + 2 -x )^2 (- 2 + 2 -x ) 1+e^ (- 2 + 2 +x ) [ _a^b f(x) d x= _a^b f(a+b-x) d x ] array = _ - 2 ^ 2 x^2 x 1+ e ^x Adding equation (i) and (ii), we get aligned & 2 I = _ - 2 ^ 2 ( x^2 x 1+ e ^x + x^2 x 1+ e ^ -x ) d x & 2 I = _ - 2 ^ 2 x^2 x [ 1 1+ e ^x + 1 1+ e ^ -x ] d x aligned aligned & 2 I = _ 2 ^ 2 x^2 x [ 1 1+ e ^x + e ^x e ^x+1 ] d x & 2 I = _ - 2 ^ 2 x^2 x ~d x aligned 2 I =2 ₀^ 2 x^2 x ~d x [ array l _ - a ^ a f (x)=2 ₀^ a f (x) d x if f (x)