If I_n= _ - ^ n x (1+ ^x ) x d x ; n=0,1,2 , , then

Step-by-step solution

(a) aligned & I_n= _ - ^ n x (1+ ^x ) x d x & I_n= _ - ^ ^x n x (1+ ^x ) x d x & [ _a^b f(x) d x= _a^b f(a+b-x) d x ] & 2 I_n= _ - ^ n x x d x aligned aligned & 2 I_n=2 ₀^ n x x d x & I_n= ₀^ n x x d x & Now, I_ n+2 -I_n & = ₀^ (n+2) x- n x x d x & = ₀^ 2 (n+1) x x x d x & =2 [ (n+1) x (n+1) ]₀^ =0 & I_ n+2 =I_n & (b) As I₃=I₅= =I₂₁ & _ m=1 ¹⁰ I_ 2 m+1 =10 I₃=10 ₀^ 3 x x d x & =10 ₀^ (3-4 ^2 x ) d x & =10[3 x-2 x+ 2 x]₀^ =10 & (c) As I₂=I₄= =I₂₀ & _ m=1 ¹⁰ I_ 2 m =10 ₀^ 2 x x d x & =20[ x]₀^ =0 & aligned