AP EAMCET201726 Apr 2017Morning ShiftMathematicsIndefinite IntegrationActual
If I_n= n x x d x , then I_n=
Options
- A-2 n-1 (n-1) x- I _ n-2
- B2 n-1 (n-1) x+ I _ n-2
- C-2 n+1 (n+1) x- I _ n-2
- D-2 n+1 (n-1) x- I _ n-2
Correct answer
A. -2 n-1 (n-1) x- I _ n-2
Step-by-step solution
aligned = & (n x-x+x) x d x = & [(n-1) x+x] x d x & (n-1) x x+ (n-1) x x d x aligned aligned & = (n-1) x d x+ (n-1) x x x d x & = (n-1) x d x+ 1 2 2 x (n-1) x x d x aligned Since, 2 x y=+ (x+y)+ (x-y) aligned & =- (n-1) x (n-1) + 1 2 n x+ (2-n) x x d x & = - (n-1) x (n-1) & + 1 2 n x x d x- 1 2 (n-2) x x d x aligned From Eq. (i), aligned & I_n=- (n-1) x (n-1) + 1 2 I_n- 1 2 I_ n-2 & (1- 1 2 ) I_n=- (n-1) x (n-1) - 1 2 I_ n-2 & I_n= -2 n-1 (n-1) x-I_ n-2 aligned