AP EAMCET202319 May 2023Morning ShiftMathematicsIndefinite IntegrationActual
1+x x x [1-x^2 (e^ x )^2 ] d x=
Options
- A1 2 | (x e^ x )^2 (x e^ x )^2+1 |+c
- B- 1 2 | (x e^ x )^2 (x e^ x )^2+1 |+c
- C1 2 | (x e^ x )^2 (x e^ x )^2-1 |+c
- D- 1 2 | (x e^ x )^2 (x e^ x )^2-1 |+c
Correct answer
C. 1 2 | (x e^ x )^2 (x e^ x )^2-1 |+c
Step-by-step solution
aligned & Let I = (1+x x) x [1-x^2 (e^ x )^2 ] & d x= (1+x x) e^ x x e^ x [1- (x e^ x )^2 ] d x aligned Let x e^ x=t e^ x (1+x x) d x=d t aligned & Hence I= d t t (1-t^2 ) = [ 1 t - 1 2(t-1) - 1 2(t+1) ] d t & I= | t |- 1 2 |( t -1)|- 1 2 |( t +1)|+C & = 1 2 | (x e^ x )^2 (x e^ x )^2-1 |+C aligned