Paragraph: Let I = ₀^ 2 f(x) g(x) d x , where f(x)= x and g(x)= x+ x+1 . Question: What is ₀^ 2 d x g(x) equal to?

Step-by-step solution

Given that g ( x )= x + x +1 aligned & ₀^ 2 d x g(x) = ₀^ 2 1 x+ x+1 d x & = ₀^ 2 1 2 2 1+ ^2 2 + 1- ^2 2 1+ ^2 2 +1 d x aligned aligned & = ₀^ 2 1+ ^2 x 2 2 x 2 +1- ^2 x 2 +1+ ^2 x 2 d x & = ₀^ 2 ^2 x 2 2 (1+ x 2 ) d x & let 1+ x 2 =t 1 2 ^2 x 2 d x=d t aligned when x = / 2 then t =2 , and when x =0 then t =1= ₁^2 1 t dt =[ t ]₁^2= 2- 1= 2