AP EAMCET20227 Jul 2022Evening ShiftMathematicsDefinite IntegrationActual
For n N , if I_n= n x x d x= 2 n-1 (n-1) x+I_ n-2 and ₀^ n x x d x= k 2 , then k=
Options
- A(-1)^n-1
- B1-(-1)^n
- C(-1)^n
- D(-1)^ n+1
Correct answer
B. 1-(-1)^n
Step-by-step solution
aligned & Consider I_n= ₀^ n x x d x, n N & I_n= [ 2 n-1 (n-1) x ]₀^n+I_ n-2 =I_ n-2 aligned aligned & I₁=I₃=I₅=I₇ . & and I₂+I₄=I₆=I₈ & Now, I₁= ₀^ x x d x= & and I₂= ₀^ 2 x d x=0 & I₁=I₃=I₅ . & = =2 . ( 2 )= (1-(-1)^n ) 2 , n odd & and I₂=I₄=I₆ . .=0 & = (1-(-1)^n ) 2 , n is even. & aligned