OCR MEI C2 (Core Mathematics 2)

Find \(\int (3x^5 + 2x^{-\frac{1}{2}}) dx\). [4]

Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}

1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]

Find \(\int \left(x - \frac{3}{x^2}\right) dx\). [3]

Find \(\int (20x^4 + 6x^{-\frac{2}{3}}) dx\). [4]

Find \(\int (12x^5 + \sqrt[5]{x} + 7) dx\). [5]

Find \(\int \left(x^{\frac{1}{2}} + \frac{6}{x^3}\right) dx\). [5]

Find \(\int \left(x^4 + \frac{1}{x^3}\right) dx\). [4]

1. Differentiate \(12\sqrt{x}\). [2]
2. Integrate \(\frac{6}{x^5}\). [3]