What is f(x) = int x+3xsqrt(x^2+1) dx if f(2) = 7 ?

1 Answer
Jul 2, 2016

\implies f(x) = x^2/2+(x^2+1)^{3/2} + 5(1- sqrt5)

Explanation:

f(x) = int x+3xsqrt(x^2+1) dx

noting the pattern D ( alpha (x^2+1)^{3/2} ) = alpha 3/2 (x^2+1)^{1/2} 2x = 3 alpha x (x^2+1)^{1/2} so here alpha = 1

\implies f(x) = x^2/2+(x^2+1)^{3/2} +C

7= 2+(5)^{3/2} +C implies C = 5(1- sqrt5)

\implies f(x) = x^2/2+(x^2+1)^{3/2} + 5(1- sqrt5)