A curve has equation y = f(x) and passes through the point (4, 22). Given that f ′(x) = 3x^2 – 3x^(1/2) – 7, use integration to find f(x), giving each term in its simplest form.

Firstly we can use the difference rule to split f'(x) into three components which we can consider separately. Then using the knowledge that the integral of x^n is 1/(n+1)*x^(n+1) we get the expression for f(x) as x^3 - 2x^(3/2) - 7x + C where C is an unknown constant.We find C by using the other information the question gives us- that when x=4, y =22. Plugging this into f(x) gives us the equation 22 = 20 +C, so C = 2. The final expression is therefore f(x) = x^3 - 2x^(3/2) - 7x + 2.