Question

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

Answer 1

`\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)`