₀^ 2 ~d x 1+( x)¹⁰¹ =

Step-by-step solution

Consider the integral I = ₀^ 2 d x 1+( x)¹⁰¹ . Applying the substitution x 2 - x yields I = ₀^ 2 d x 1+( x)¹⁰¹ . Adding these two expressions gives 2I = ₀^ 2 ( 1 1+( x)¹⁰¹ + 1 1+( x)¹⁰¹ ) d x . The integrand simplifies to 1 since 1 1+( x)¹⁰¹ + 1 1+( x)¹⁰¹ = ( x)¹⁰¹ ( x)¹⁰¹+1 + 1 1+( x)¹⁰¹ = 1 . Therefore, 2I = ₀^ 2 1 d x = 2 , and I = 4 . Final answer: I = 4