Paragraph: Let P= ₀^1 x 1-x ( x 1-x ) d x, Q= ( +1 +1 2 ) and R= ₀^8 e^ Q / P d . Question: The value of 3 R is equal to:

Step-by-step solution

P= ₀^1 x 1-x ( x 1-x ) d x Put aligned & x 1-x =t^2 x=t^2-t^2 x & x= t^2 1+t^2 =1- 1 1+t^2 & d x= 2 t (1+t^2 )^2 & P= ₀^ t(2 t) 2 t (1+t^2 )^2 d t & P=2 ₀^ (t t) _ (i) 2 t (1+t^2 )^2 _ (ii) d t & aligned aligned P & =2 [ .(t t) ( -1 1+t^2 ) |₀ ^ + ₀^ (1+ t) 1+t^2 d t ] & =2[0+ ⁻¹ t ₀]₀^ + ₀^ t 1+t^2 d t ₀ & = Now, R & = ₀^8 +1 +1 2 d_ = 38 3 3 R & =38 aligned