Integration

(a) Use the substitution u = 1 + x² to show that ∫₀¹ 4x/((1 + x²)²) dx = 1. (b) Use integration by parts to find the exact value of ∫₁^e x² ln x dx.

Water drains from a tank. The volume of water, V litres, after t minutes is modelled by dV/dt = -k√V, where k is a positive constant. Initially V = 100. After 10 minutes, V = 64. (a) Solve the differential equation to find V in terms of t. (b) Find the time taken for the tank to empty. (c) Give one reason why the model may not be accurate as the tank nearly empties.

(a) Express (5x + 4)/((x + 2)(x - 1)) in partial fractions. (b) Hence find ∫ (5x + 4)/((x + 2)(x - 1)) dx.

(a) The curve y = x² - 4x + 5 and the line y = x + 1 intersect at two points. Find the exact area of the region enclosed between them. (b) Use the trapezium rule with 4 strips to estimate ∫₀² √(1 + x³) dx, and state whether your answer is an overestimate or an underestimate, giving a reason.

(a) Find the exact value of ∫₀^π/2 sin² x dx. (b) Find ∫ 6x/(3x² + 1) dx. (c) A curve is given parametrically by x = t², y = t³ for t ≥ 0. Find the area of the region bounded by the curve, the x-axis and the line x = 4.