Use the formula ∫ 𝑥𝑛 𝑑𝑥 = 𝑥𝑛+1 𝑛+1 + 𝑐 for 𝑛 ≠ − 1.

A particle moves along a straight line such that its acceleration at time \(t\) seconds is given by \(a(t) = 12t^2 - 8t\) metres per second squared. At \(t = 0\), the particle has a velocity of \(5\) m/s and is located \(2\) metres from the origin. (a) Use anti-differentiation to determine the velocity function \(v(t)\). (2 marks) (b) Use the result from part (a) to determine the position of the particle at \(t = 3\) seconds. (2 marks)

A scientist models the rate of change of the volume of a chemical reaction vessel using the function \(V'(t) = 12t^2 - 8t + 3\), where \(V'(t)\) is measured in litres per minute and \(t\) is the time in minutes after the reaction begins. (a) Find the antiderivative \(V(t)\). (1 mark) (b) Given that at \(t = 0\) the volume is 50 litres, determine the constant of integration. (1 mark) (c) Calculate the volume of the vessel at \(t = 2\) minutes. (1 mark)

Find the antiderivative of \(3x^5 - 4x^3 + 7\).

Evaluate \(\displaystyle \int (3x^4 - 5x^2 + 2) \, dx\).