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 intergration to find f(x).

As f '(x) is the differential of f(x) we have to first integrate f '(x). To do this we take each term individually and integrate it. So starting with 3x², to integrate a simple function of x like this we have to add 1 to the power and then divide by the new power so 3x² integrates to x³. Then we do the other two terms in the same way so 3x^0.5 integrates to 2x^1.5 and note that 7 is the same as 7x⁰ so we add one to the power and so then divide by 1 so 7 integrates to 7x. So f(x) = x³ - 2x^1.5 - 7x + c It's important to remember to add the constant of integration because without this it is wrong and you will lose marks. Next we're going to find the constant of integration and to do this we substitute the x,y values they have given us in the question into our equation. So f(4) = y = 22 = 4³ - 24^1.5 - 74 + c So c = 2 Then f(x) = x³ - 2x^1.5 - 7x + 2 and we can't simplify it any further so the question is finished.